Problem Set 2
1. A is a non-empty set. Suppose ¯a= supA and a = infA.
(i) Take α >0 and define C ={αx |x ∈A}. Show that, supC =α¯aand infC =αa.
(ii) Take α <0 and define C ={αx|x ∈A}. Show that, infC =α¯aand supC=αa.
2. A and B are non-empty sets such that a ≤ b for all a ∈ A and b ∈ B.
Show that, supA≤infB.
3.LetAandBbe non-empty bounded sets withA⊆B. Show that supA≤ supB and infA≥infB.
4.Prove or provide counterexample
(i) A finite non-empty set always has maximum and minimum.
(ii) A and B are non-empty sets such that a < b for all a∈A and b ∈B.
Then supA <infB.
(iii) Ifa= supA then we can find a sequence{an}∞n=0 such thatan∈Afor alln and liman=a.
5.Let liman=aand limbn=b
(i) Take the sequence{anbn}∞n=0. Show that limanbn=ab.
(ii) Supposebn6= 0 for all nand b6= 0. Take the sequence n
an
bn
o∞
n=0. Show that lim
an
bn
= a b.
6.Find the limit of following sequence:
(i) an= 2n−3n+3, (ii) an= 2nn, (iii)an= 1n(−1)n
(iv) a1= 0 and an+1 = 10−a2 n
(v) 1,0,1,0,0,1,0,0,0,1,0,0,0,0,1,(5 zeros),1, . . .
7. (Squeeze Theorem) If xn ≤ yn ≤ zn for all n and limxn = limzn = L then limyn=L.
8.Prove or provide counterexample
(i) If a sequence is not bounded then it does not have converging subse- quence.
(ii) If a monotone sequence diverges then it does not have converging sub- sequence.
(iii) If a sequence {an}∞n=0 converges then for every > 0, there exists N such that|am−an|< ∀m, n≥N.
(iv) If{an}∞n=0 and {bn}∞n=0 both diverge then{(an+bn)}∞n=0 also diverge.
(v) IfPn
i=1ai= 1 then Pn
i=1a2i ≥ n1.
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