Important Proofs you should know · 2020-06-07 · (1)Triangle Inequality and Reverse Triangle Inequality (2)max(S) and min(S) (3) Mis an upper bound for S, mis a lower bound for

MATH 140A − FINAL EXAM − STUDY GUIDE

The Final Exam takes place on Tuesday, June 9, 2020 from 1:30pm to 3:50 pm. It is a closed book and closed notes exam, and countsfor 30% of your grade. You will take the exam on Canvas, the sameway you take your quizzes. Note that the 140 mins to take the examincludes the time to upload your answers. If you finish your exambetween 3:45 pm and 3:50 pm, please e-mail it to me instead of up-loading it to Canvas. I will NOT accept any submissions after 3:50pm. There will be 8 problems in total, so you should roughly spend15 mins per problem. The last problem is harder than the rest.

The Final Exam is cumulative and covers everything from section 1 upto and including section 19, except for sections 6 and 16 (which willnot be on the exam). 3−4 questions will come from the material afterthe second midterm (sections 15 though 19) and 1 − 2 questions arefrom section 13 (metric spaces), so make sure to study those sectionsparticularly well.

Important Proofs you should know

Know how to prove the following theorems. I will not explicitly askyou to reprove them, but I could give you variations of them, or askabout proving parts of the theorems.

(1) inf(S) = − sup(−S) (section 4) and lim inf sn = − (lim sup−sn)

Date: Tuesday, June 9, 2020.

1

2 MATH 140A − FINAL EXAM − STUDY GUIDE

(2) Archimedean Property

(3) Any of the 10 Examples of Limits in Lectures 7-9 (see sections8 and 9 below)

(4) (sn) converges⇒ (sn) is bounded (section 9), and similarly (sn)is Cauchy ⇒ (sn) is bounded

(5) Limit laws such as sn + tn → s+ t or sntn → st or tnsn→ t

s

(6) Monotone Sequence Theorem

(7) E is closed if and only if Ec is open

(8) If E is compact, then E is closed and bounded

(9) Divergence Test

(10) Root Test (just the case α < 1)

(11) Integral Test

(12) The sequence ε−δ definitions of continuity are equivalent (The-orem 17.2)

(13) Any of the 7 Examples of Continuity covered in lecture and/orthe HW (see section 17 below)

(14) f + g, kf , |f |, fg, fg , g ◦ f are continuous (both using sequences

and ε− δ)

(15) Extreme Value Theorem (Theorem 18.1; including the state-ment about f bounded)

(16) Intermediate Value Theorem

MATH 140A − FINAL EXAM − STUDY GUIDE 3

(17) Fixed Points (Example 1 in section 18) and Square Roots (Ex-ample 2 in section 18)

(18) Any of the 3 examples in Lecture 27 to show that f is (or isnot) uniformly continuous

(19) f continuous on [a, b] ⇒ f is uniformly continuous

(20) f ′ bounded ⇒ f uniformly continuous (Theorem 19.6)

Concepts you should know

I will not explicitly ask you for definitions, but know the following con-cepts.

(1) Triangle Inequality and Reverse Triangle Inequality

(2) max(S) and min(S)

(3) M is an upper bound for S, m is a lower bound for S

(4) sup(S) = M , inf(S) = m

(5) Least Upper Bound Property

(6) sup(S) =∞, inf(S) = −∞

(7) Archimedean Property

(8) Q is dense in R (also know that for any a ∈ R there is a sequence(sn) in Q converging to a)

(9) (sn) converges to s (both in R and metric spaces)


(10) The Squeeze Theorem

(11) sn →∞, sn → −∞

(12) Increasing/Decreasing (both for sequences and functions)

(13) lim sup sn and lim inf sn

(14) lim sup squeeze theorem

(15) Cauchy Sequence (both in R and in metric spaces)

(16) Completeness (both in R and in metric spaces)

(17) Subsequence

(18) Limit Point

(19) Bolzano-Weierstraß Theorem (both in R and Rk)

(20) Metric Space

(21) B(x, r), open set, interior point, E◦

(22) Closed set, limit point, E

(23) Boundary ∂E

(24) Finite Intersection Property

(25) Cantor Set

(26) Compactness, Open cover, sub-cover, finite sub-cover

(27) E is bounded

(28) Heine-Borel Theorem


(29) Series, partial sum,∑an converges/diverges

(30) Geometric Series∑rn

(31) Cauchy Criterion

(32) Divergence Test

(33) Comparison Test

(34) Absolute Convergence

(35) Root Test

(36) Ratio Test

(37) Integral Test

(38) Alternating Series Test

(39) f continuous (both the ε− δ and the sequence definition)

(40) Extreme Value Theorem (Theorem 18.1)

(41) Intermediate Value Theorem

(42) f(I)

(43) Uniformly Continuous

(44) Continuous Extensions

(45) All the tests for uniform continuity:

(i) f continuous on [a, b] is uniformly continuous

(ii) f is uniformly continuous iff f has a continuous extension


(iii) f f ′ is bounded, then f is uniformly continuous

(iv) If (sn) is Cauchy and f is uniformly continuous, then f(sn)is Cauchy

(v) If S is bounded and f is uniformly continuous on S, thenf is bounded (problem 19.4)

Section 1: The set N of natural numbers

• Don’t spend too much on this section, all you need to knwois how to prove a statement by induction. Examples 1 and3, Problems 1.9 and AP1 and AP2 in HW1 are good practiceproblems.

• Also check out AP3 in HW1, it’s a good “Find a counterexam-ple” exercise

Section 2: The set Q of rational numbers

• Don’t spend too much time on this section. Pretty much allyou need to know is how to show that

√2 is irrational. It’s a

classical proof that is hopefully familiar to you from previouscourses.

• IGNORE everything that has to do with the rational rootstheorem

• You can ignore AP4-AP6 in HW1. In particular, I won’t askabout equivalence relations on this midterm


Section 3: The set R of real numbers

• Don’t spend too much time on this section. The most importantthing here is the Triangle inequality.

• Understand the proof of the corollary to the triangle inequal-ity (with dist(a, b)), it illustrates an important technique that’sused over and over again

• Know the reverse triangle inequality (Problem 3.5)

Section 4: The Completeness Axiom

• This is the most important section from the Midterm 1 material.

• Define the concept of max and min and show that S has a maxor doesn’t have a max or a min. The examples on page 2-5 ofLecture 4 are excellent practice examples

• Know how to show (or not) that S is bounded above (or below)

• Define sup(S) and inf(S) and show that sup(S) = M (or inf(S) =m). The examples on page 9-11 and 13-15 of Lecture 4, as wellas AP3 in HW 2 are excellent practice with that.

• Know the least upper bound property

• Show that inf(S) = − sup(−S) and to deduce the greatest lowerbound property from that (page 6-7 of Lecture 5)

• Know the statement and the proof of the Archimedean propertyand know how to use it


• Know the statement of Q dense in R. You don’t need to mem-orize the proof, but definitely understand it. In particular, notehow the Archimedean property is used here

• Also check out Problems 4.7, 4.8, 4.14 (important), 4.16 andAP3 in HW 2. If you want more practice, also check out 4.10and 4.15

• Ignore AP5 and AP6 in HW 3

Section 5: The Symbols ∞ and −∞This section is super short. Just know that sup(S) = ∞ means S isnot bounded above and inf(S) = −∞ means S is not bounded below.If you want more practice, check out AP3(c) from HW2 or check out5.2 (prove those statements)

Section 6: A Development of RIGNORE this section, it will NOT be on the exam. One might evensay I cut it out from the exam material ,

Section 7: Limits of Sequences

The only important thing in that section is the definition of a limit(Definition 7.1 or page 5 in Lecture 7) and the fact that limits areunique (pages 3-4 in Lecture 8). You don’t need to know the definitionof a sequence. But check out Problem 7.4, it’s neat!

Section 8: A Discussion about Proofs

• This section is also important.


• Know how to do ALL the examples in this section and thelectures, they are all fair game and good practice with the def-inition of a limit. The examples in lecture include

- Example 1: The Basics, sn = 3− 1n2

- Example 2: Simple Fraction, sn = 2n+44n+5

- Example 3: Complex Fraction, sn = 2n3+3nn3−2

- Example 4: The Limit Does Not Exist, sn = (−1)n

- Example 5: Square roots, sn → s⇒ √sn →√s

- Example 6: sn → s⇒ |sn| → |s| (see AP3 in HW 3)

• Note: It’s important to write down BOTH the scratch workand the actual proof, otherwise you WILL lose points!

• No need to know how to prove the statement about sequencesthat are bounded away from 0 (pages 6-9 of Lecture 8)

• Know the statement and the proof of the Squeeze Theorem (Seeproblem 8.5)

• Of course, problems 8.1, 8.2, 8.3, 8.4, 8.7, 8.8, 8.9, and 8.10 areexcellent practice problems

Section 9: Limit Laws for Sequences

• Prove limit laws such as

- If sn → s and tn → t, then sn + tn → s + t (Page 10-11 ofLecture 8)


- If sn → s and tn → t then sntn → st (Page 1-2 of Lecture9)

- If sn 6= 0 and sn → s 6= 0 and tn → t, then tnsn→ t

s

(Pages 2-4 of Lecture 9); you’d of course have to show theintermediate step of Example 7

• Know how to show that if (sn) converges, then (sn) is bounded(Last section in Lecture 8)

• Know how to show

- Example 7: If sn 6= 0 and sn → s 6= 0, then 1sn→ 1

s

- Example 8: If |a| < 1, then limn→∞ an = 0

- Example 9: limn→∞ n1n = 1 and its corollary limn→∞ a

1n = 1

if a > 0

- Example 10: Infinite limits such as limn→∞√n− 2+3 =∞

• Note: Also know how to show that limn→∞1np = 0 if p > 0

(This is just 8.1(d) but with p instead of 3)

• Know the binomial theorem (but you don’t need to know howto prove it)

• Define limn→∞ sn =∞ and limn→∞ sn = −∞ and know how toshow that a sequence goes to ∞ (like Example 10 above)

• Prove some limit laws for infinite limits, such as if sn →∞ andtn →∞, then sntn →∞ (page 11 of Lecture 9) if sn →∞ andtn ≥ m for some m, then sn + tn →∞ (Problem 9.11(c))

• Know the Duality Formula (last section of Lecture 9), and useit to show for example that limn→∞ 2n =∞


• Problems 9.6(a)(b), 9.9, 9.10, 9.11, 9.17, AP1, and AP2 in HW4 are great practice problems

• Ignore 9.12 and 9.15 (they’re on HW 4 though) and ignore AP3 on HW 4.

Section 10: Monotone Sequences and Cauchy

Sequences

• Define: increasing/decreasing sequences

• Prove the Monotone Sequence Theorem (Theorem 10.2), as wellas its corollaries: “If (sn) is decreasing and bounded below, thenit converges”, and “If (sn) is increasing, then it either convergesor goes to ∞

• Ignore the discussion about Decimal Expansions

• Define: lim supn→∞ sn and explain why lim sup exists (basi-cally because sup {sn | n > N} is decreasing and bounded be-low); same with lim inf

• Find, with proof, lim supn→∞(−1)n and lim inf(−1)n

• Prove lim inf sn = − (lim sup−sn). This is an important iden-tity that allows us to go from lim sup to lim inf

• Two important facts about lim sup (no need to know the proof)

(1) If sn → s, then lim inf sn = lim sup sn = s

(2) Limsup squeeze theorem: If lim inf sn = lim sup sn = s,then sn → s


• Define: Cauchy sequence, it’s important

• Prove: (sn) is converges ⇒ (sn) Cauchy

• Prove: Cauchy sequences are bounded

• Know the fact that Cauchy sequences in R are convergent, butyou don’t need to know the proof

• Define: completeness and convince yourself that Q is not com-plete

• In HW 5, I recommend checking out AP1, AP2, AP3, and AP4,and 10.6, but you can ignore AP5, the optional AP’s, and 10.8

Section 11: Subsequences

• Understand the concept of a subsequence, but you don’t needto know the definition with σ.

• Remember that nk ≥ k (the express train is faster than theoriginal train)

• Prove that if sn converges to s, then every subsequence snk con-verges to s

• Know how to do an inductive construction. The followingexamples are good practice with this concept:

(1) If (rn) is an enumeration of the rational numbers and a ∈ R,then there is a subsequence rnk that converges to a

(2) If (sn) is a sequence of positive numbers with inf {sn | n ∈ N} =0, then there is a decreasing subsequence snk that convergesto 0.


(3) The first problem on the mock midterm 2

(4) Problem 10.7

(5) The first question on Midterm 2

• Fact: every sequence has a monotonic subsequence, but ignoreits proof, although it is pretty neat!

• Fact: there is a monotonic subsequence of (sn) that converges tolim supn→∞ sn, but please ignore its proof, for your own sanity,

• That said, it is easier (and a good exercise with the inductiveconstruction) to show that there is a subsequence (not neces-sarily monotonic) of (sn) that converges to lim supn→∞ sn; tryit out if you want

• Define: Limit point, and find limit points of sequences such assn = (−1)n or sn = sin

(πn3

)or sn = 0 if n is even and n if n is

odd Know (and know how to prove) the following facts: If S isthe set of limit points of (sn), then

(1) S 6= ∅(2) sup(S) = lim sup sn

(3) (sn) converges if and only if S is a single point

• Know the Bolzano-Weierstraß Theorem, but no need to knowhow to prove it.

• Fact: set of limit points of (sn) is closed, but no need to knowthe proof

• In HW5, 10.6 is a good practice with the inductive construction,and also check out 11.8 and 11.9, as well as AP6 and AP7


Section 12: lim sup and lim inf

• Show that, in general, lim sup sntn 6= (lim sup sn) (lim sup tn)

• Show that if sn → s > 0 then lim sup sntn = (lim sup sn) (lim sup tn),but only do it in the case where lim sup tn is finite

• Do NOT memorize the pre-ration test, but understand themain ideas of the proof of the pre-ratio test. This proof isused again in section 14 to prove the root test. In particular,understand why the pre-ratio test shows that the root test isbetter than the ratio test

• In HW6, check out AP1 and AP6, as well as 12.4, 12.5, and12.8. You can ignore 12.12 and 12.13 if you want.

Section 13: Some Topological Concepts in

Metric Spaces

Note: You are guaranteed to have 1 − 2 questions on the final onmetric spaces. The last question on the final (which is the hardest)will involve compactness while the other 7 will not, so it’s up to you ifyou want to study compactness or not.

• Know the definition of a metric space

• You don’t need to memorize Examples 1-10, but understandwhat they’re saying. I will define the space for you and themetric if necessary (just like the third example on the practicemidterm) The only exception is Example 5 in the notes; youneed to know what the discrete metric is, since it’s a great sourceof counterexamples


• Define: convergence in a metric space

• Know the notation used for points and sequences in Rk. Inparticular, (x1, . . . , xk) is a point in Rk but (x(n)) is a sequencein Rk

• Fact: A sequence in Rk converges if and only if every componentconverges. For the proof, you only need to know the ⇒ part.

• Don’t memorize the Squeeze Theorem for distances; you onlyneed to know that |xj − yj| ≤ d(x, y)

• Define: Cauchy sequence in a metric space, and complete metricspace

• Show that Rk is complete

• Know the Bolzano-Weierstraß Theorem in Rk and know how toshow it

• Define: open ball B(x, r) and open set

• Remember that in R, B(x, r) = (x− r, x+ r)

• Show that (a, b) is open and [a, b] is not open

• Show that the union of any collection of open sets is open, andthe intersection of finitely many open sets is open

• Define an interior point, and E◦

• Find E◦ for various examples, such as [0, 1], Q, (0, 1),{

1n , n ∈ N

},

the Cantor set

• Know that E is open if and only if E◦ = E


• Define: E is closed. Please use the definition given in lecture;the one in the book is a bit awkward

• Show (0, 1) is not closed, but that [0, 1] is closed

• Show E is closed if and only if Ec is open

• Know that the intersection of any number of closed sets is closed,and the union of finitely many closed sets is closed, and knowhow to prove this

• Define: Limit point of E, and E (the book uses E−)

• Find E for various examples, such as (0, 1), [0, 1], Q, B(x, r),{1n , n ∈ N

}• Know that E is closed if and only if E = E

• Show Ec = (E◦)c (see practice exam)

• Define ∂E (boundary of E) and find ∂E for various examples,such as ∂[0, 1] or ∂B(x, r) (in Rk) or ∂Q or ∂

{1n , n ∈ N

}• Also see Question 3 on Midterm 2.

• State and prove the Finite Intersection Property, and give acounterexample if the sets are not closed

• Understand the construction of the Cantor set and show that itis closed, nonempty, has size 0, empty interior, and compact

• Ignore the discussion on decimal expansions and the Cantor set

• Define: U is an open cover of E, V is a subcover of U , V is afinite subcover of U


• Define: E is compact

• Show R and (0, 1) are not compact

• Define: E is bounded

• Show that E is compact, then E is closed and bounded

• Notice in particular that, for finite sets, you can define thingslike max {n1, . . . , nk} (as in the proof above), which you cannotdo for infinite sets! That’s why finite subcovers are important

• Know the statement of the Heine-Borel Theorem, but ignore itsproof

• In the second edition of the book, ignore Example 6 with d(E, x)(which isn’t part of the first edition)

• In HW6, know how to do 13.1, 13.4, and 13.10 but please ignore13.7. I’d also highly recommend looking at 13.9. You can ignoreAP2, although it’s good practice with balls, but check out AP3,AP4, and AP5, but ignore AP 7−9. Note: In AP5, know thedefinition of dense

• In HW7, know how to do 13.12 and 13.15(a). You don’t needto know how to do 13.15(b), except for the end where, once youproved that distinct x(n) cannot belong to the same U(x), thenyou find a contradiction. Also know how to do AP1, AP2, andAP3, but you don’t need to know the definition of sequentiallycompact or totally bounded. Ignore AP8 and AP9

• Also check out Problem 4 on Midterm 2.


Section 14: Series

• Know the definition of a series and the definition of partial sums

• Define: A series converges, a series diverges

• You don’t need to know that 1n(n+1) = 1

n −1

n+1

• Show that if an ≥ 0, then∑an converges if and only if (sn) is

bounded

• Rigorously find the value of the geometric series∑rn and show

that it converges if and only if |r| < 1

• Derive the Cauchy criterion for a series and know how to defineit

• Prove the Divergence test: If∑an converges, then an → 0

• State and prove the Comparison Tests

• Define:∑an converges absolutely, and show that absolute con-

vergence ⇒ convergence

• State the Root Test and know how to prove it in the case α < 1

• State the Ratio Test; I will not ask you to reprove it.

• Of course, all the examples in the lecture and in the book (es-pecially Examples 3−9 in the book) are fair game for the exam

• In HW7, check out 14.6a, but ignore 14.10, and check out AP4,AP5 (with the hints), AP6 (but don’t memorize the block test).Ignore AP7, AP9, AP10. The Binomial Theorem is fair gamefor the final exam.


Section 15: Alternating Series and Integral

Tests

• Prove the Integral Test (both versions). I could ask you to proveit for a specific example, like Examples 1 and 2 in the book, orI could ask you to do it in general (like AP1 on HW8)

• Prove the important fact that∑

1np converges if and only if

p > 1

• Know the statement of the alternating series test (and how touse it), but you don’t need to know its proof

• In HW9, check out problem 15.6 (it’s a great problem in testingyour understanding of series), ignore the Cauchy-Schwarz in-equality AP2, know how to do AP3, and ignore problems AP4-AP6 and all the optional APs

Section 16: Decimal Expansions of Real

Numbers

Ignore this section (for your own sanity), it will not be on the exam

Section 17: Continuity

• Know both definitions of continuous functions (with sequencesand ε − δ). Seriously, if you ever want to get a tattoo (whichI don’t necessarily recommend), you should choose the ε − δdefinition as your design (although it’s a bit long, so it mighthurt ,). This definition will be used over and over again inanalysis.


• Show that the ε − δ definition and the sequence definition areequivalent (Theorem 17.2), it’s a very classical proof that usesideas that we’ve talked about before

• Use both the ε−δ definition and the sequence definition to showthat some familiar functions are continuous:

- Example 1: Basic Functions like f(x) = x2 + 1

- Example 2: x2 sin(1x

)at x = 0 (using ε− δ and sequences),

1x sin

(1x2

)(using sequences)

- Example 3: Not continuous (see Lecture 23 and also Prob-lems 17.10(a) and 17.10(b))

- Example 4: x3 (Problem 17.9(d))

- Example 5: |x| (AP1(a) in HW9)

- Example 6: 1x (AP1(b) in HW9)

- Example 7:√x (AP1(c) in HW9)

- Lipschitz Functions (AP2 in HW9)

• Show that if f and g are continuous, then f + g, kf , f − g, |f |,fg, 1

f , fg and g ◦ f are continuous. You need to know how to do

that both with the sequence definition and the ε− δ definition

• You don’t need to know the formula for max(f, g) in Example8; I would give you the formula if needs be

• Notice in particular how the idea of Contradiction + Bolzano-Weierstraß appears in so many proofs in 140A!

• In HW 9, ignore AP3 and AP4 and all the optional APs, butknow how to do 17.12, 17.13, and 17.14


Section 18: Properties of Continuity

• Show that continuous functions on [a, b] are bounded (see Lec-ture 25) It’s a very classical proof.

• In particular, notice how we use the idea of Contradiction +Bolzano-Weierstraß . This idea appears over and over again in140A and 140B

• Prove the Extreme Value Theorem. I know it looks long, butthe main idea is to find a sequence that converges to the sup off

• Prove the Intermediate Value Theorem. The main idea is toshow that there is x such that f(x) ≤ c and f(x) ≥ c

• Know the statement that if f is continuous and I is an interval,then f(I) is an interval, but ignore its proof

• Beware that if I is an open set, then f(I) might not be an openset. For example, if f(x) = x2, then f((−1, 2)) = [0, 4)

• Know how to prove some applications of the Intermediate ValueTheorem: For example, if f : [0, 1] → [0, 1] is continuous, thenf has at least one fixed point (Example 1 in the book), or thatfor every y ≥ 0 there is some x ≥ 0 such that x2 = y (Example2)

• Important: Even though the section on√x was marked op-

tional at the time, you should still know it, and I could still askyou about it

• Know the fact that one-to-one continuous functions are eitherstrictly increasing or strictly decreasing, but ignore its proof


• Know that if f is one-to-one and continuous, then f−1 is con-tinuous, but ignore its proof

• That said, know how to show that if f is increasing, then f−1

is increasing (see page 18 in Lecture 26)

• In HW10, know how to do 18.9, 18.10, and 18.12, as well asAP1

Section 19: Uniform Continuity

• Know the definition of uniform continuity, and understand howit’s different from the definition of continuity. The main pointis that, in uniform continuity, the δ does not depend on x andy, it’s independent of your position

• Use the ε − δ definition of uniform continuity to show that fis uniformly continuous. For example, show that f(x) = x2 isuniformly continuous on [−1, 3] (Example 1 in Lecture 27) orthat f(x) = 1

x2 is uniformly continuous on [2,∞) (Example 2 inLecture 27)

• Use the ε− δ definition of uniform continuity to show that f isNOT uniformly continuous. For example, show that f(x) = 1

xis not uniformly continuous on (0, 1) (Example 3 in Lecture27). The book gives a very strange way of showing this, so I’drecommend you to check out the lecture notes instead

• Show that if f is continuous on [a, b], then f is uniformly con-tinuous on [a, b]. Again, the usual Bolzano-Weierstraßtrick. Usethis, for example, to show that f(x) = x2 is uniformly continu-ous on [−1, 3]


• Show that if sn is Cauchy and f is uniformly continuous, thenf(sn) is Cauchy. Use this to show that f(x) = 1

x is not uniformlycontinuous on (0, 1)

• Know the definition of continuous extension

• Prove that if f : (a, b) → R has a continuous extension f :[a, b]→ R, then f is uniformly continuous on (a, b). Use this toshow that f(x) = x sin

(1x

)is uniformly continuous on (0, 1]

• Know that if f : (a, b) → R is uniformly continuous on (a, b),then f has a continuous extension f : [a, b]→ R, but you don’tneed to prove this. Use this to show that f(x) = sin

(1x

)is not

uniformly continuous on (0, 1]

• Ignore the example with f(x) = sin(x)x

• Know the statement of the Mean Value Theorem (see Lecture28), but you don’t need to know its proof

• Prove that if f : [a, b] → R and if f ′ is bounded on (a, b), thenf is uniformly continuous. Use this to show that f(x) = 1

x isuniformly continuous on [2,∞).

• In the book, you can check out Example 1 (which is just practicewith continuity), but ignore Discussion 19.3

• In HW10, problems 19.2, 19.4 − 19.8 are good practice withuniform continuity, but you can ignore 19.9 and 19.11.

Documents

Important Proofs you should know · 2020-06-07 · (1)Triangle Inequality and Reverse Triangle Inequality (2)max(S) and min(S) (3) Mis an upper bound for S, mis a lower bound for