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...