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Math 300, Spring 09: Midterm Solution
March 3, 2008
1. (10 points). Let {an} be a sequence in R such that for some 0 < r < 1, |an+1−an| < rn,
for all n ≥ 1. Prove that the sequence {an} converges. You can assume
nsummationdisplay
k=0
rk = 1−r
n+1
1−r , and
∞summationdisplay
k=0
rk = 11−r if |r| < 1.
If you use any important theorems, STATE (without proof) those theorems.
Solution: To show {an} is Cauchy: For any n ≥ m,
|an −am|≤|an −an−1|+ ... +|am+1 −am| < rn−1 + ... + rm
<
∞summationdisplay
i=0
ri −
m−1summationdisplay
i=0
ri = 11−r − 1−r
m
1−r =
rm
1−r.
Fix any epsilon1 > 0, Choose N large enough such that rN1−r < epsilon1. Hence for all n ≥ m ≥ N,
|an −am|≤ r
m
1−r ≤
rN
1−r < epsilon1.
This shows {an} is Cauchy. But every Cauchy sequence converges in R. Hence {an}
converges.
2. (10 points)
(a) (2 points) State the Bolzano-Weierstrass Theorem in R: Every bounded sequence
in R has a convergent subsequence in R.
(b...

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