Math 521, Lecture 2 - Spring 2009
Advanced Calculus
Homework 4
Exercise 1
Let k∈]0,1[.Let (xn)be a sequence satisfying for every n∈N
|xn+2 −xn+1| ≤ k|xn+1 −xn|.
(1) Prove that for every n∈N:
|xn+2 −xn+1| ≤ kn+1 |x1−x0|.
(2) Prove that if p, q are integers such that p > q ≥0, then:
p−1
X
i=q
ki=kq1−kp−q
1−k≤kq
1−k.
(3) Prove that if p, q are integers such that p≥q≥0, then:
|xp−xq| ≤ kq|x1−x0|
1−k.
(4) Prove that (xn)is a Cauchy sequence.
Exercise 2 Let An⊂Rbe a set for every n∈N. We define ∩n∈NAnas follows: x∈ ∩n∈NAnif and
only if for every n∈Nx∈An. We define ∪n∈NAnas follows: x∈ ∪n∈NAnif and only if there is an
integer n∈Nsuch that x∈An.
(1) Let An= (−1
n,1
n). Prove that ∩n∈NAn={0}.
(2) Let An= [ 1
n,1−1
n]. Prove that ∪n∈NAn= (0,1).
Exercise 3 Let (xn),(yn)be two sequences satisfying xn< ynfor every n∈N. Set In:= [xn, yn].
Assume that In+1 ⊂Inand that the sequence (yn−xn)converges to 0.
We assume first that α:= sup{xn,n ∈N}exists.
(1) Prove that for every p, q ∈N,xp≤yq.
(2) Prove that for every q∈N,α≤yq.
(3) Prove that α∈ ∩n∈NIn.
(4) Prove that ∩n∈NIn={α}(clue: assume by contradiction that there is α06=αand estimate |α−α0|
using |xn−yn|).
We don’t assume anymore that sup{xn, n ∈N}exists. But we assume that ∩n∈NIn={c}.
(5) Prove that sup{xn, n ∈N}exists and that c= sup{xn, n ∈N}.
Honour Exercise Using the previous exercise, we want to prove that Ris uncountable. So assume by
contradiction that Ris countable.
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