Question 7

Statement. II lim an) = lal, then lim a, = a? 12 - +00 n - + 00 Question 5. (Total 10pts) Prove the Monotone Convergence Theorem: Each bounded and monotone sequence will converge. Question 6. (Total 12pts) Prove that the sequence } V2, V2 + V2, 2 + V2 + V2, 2+ 2+ V2 + v2, ... > is bounded and increasing, so that it converges; find its limit as well. Question 7. (Total 12pts) Prove the Bolzano-Weierstrass Theorem: Each bounded sequence will have a convergent subsequence. Question 8. (Total 12pts) Recall the concept of a Cauchy sequence, make a comparison with the &, & formulation of a convergent sequence, and prove that a sequence converges if and only if it is a Cauchy sequence. Question 9. (Total 10pts) Provide a formula of Cn in terms of an, bn such that End, Cn = (n-1 an) (Eng, bn). Question 10. (Total 8pts) Find a double-indexed sequence amn such that a bizarre phenomenon _m=1 2n=1 amn = 0 while to to n=1 Lm=1amn = 2 will occur