ASSIGNMENT 1

Q1:- Prove that there is no rational number whose square is 12.Q2:- Let S be a non-empty subset of real numbers, bounded above.

Show that if u = sup S, then for every natural number n, the numberu − 1

nis not an upper bound of S, but the number u + 1

nis an upper

bound of S.Q3:-Let A and B be bounded subsets of R, Then show that A ∪ B

is a bounded subset of R. Show that

sup A ∪B = max {sup A, sup B} .

Q4:- Prove that no order can be de�ned in the complex �eld thatturns it into and ordered �eld.

Q5:- Let A be a non-empty set real numbers which is bounded below.Let −A be the set of numbers −x, where x ∈ A. Prove that

inf (A) = − sup (−A) .

Q6:- Let A and B be non-empty subsets of R and let A ⊆ B. Thenshow that

inf (B) ≤ inf (A) ≤ sup (A) ≤ sup (B) .

Q7:- Find Supremum and In�mum of the following sets:-

(1) sup{1− 1

n: n ∈ N

}and inf

{1− 1

n: n ∈ N

}.

(2) sup (Q) and inf (Q) .(3) sup

{n− 1

n: n ∈ N

}and inf

{n− 1

n: n ∈ N

}.

(4) sup {x ∈ Q : x2 < 2} and inf {x ∈ Q : x2 < 2} .

Q8:- Find the Supremum and In�mum of the set A ={

mm+n

: m, n ∈ N}

.Q9:- Determine whether the following sets are bounded (from below,

above or both). If so, determine their in�mum and / or supremum and�nd out whether these in�ma/sprema are actually minima/maxima.

(1) A = {1 + (−1)n : n ∈ N} .(2) A =

{1m

+ 1n

: m, n ∈ N}

.(3) A = {x ∈ R : x2 + x + 1 ≥ 0} .(4) A =

{cos

(nπ3

): n ∈ N

}.

Q10:- Verify the following by induction.

(1 + x)n ≥ 1 + nx x > −1

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