Cauchy sequence

[Grant Bonner]

This is a three part question:

1. Let {a_n} be a bounded sequence. For every n in J, define y_n= sup{a_k: k>=n}
a. Prove that {y_n} is decreasing and bounded from below.
b. Let lim {y_n} -> A. Prove that {a_n} has a subsequence that converges to A.

2. Using item 1, prove every Cauchy sequence is convergent.

Im assuming using part a we want to show that there is some m, such that a_n <= a_m <= y_n. And from there show that |a_n - a_m| < e. My issue is with part a. I can see that {y_n} must be bounded below by {a_n}, because {y_n}>= max{a_n}.

 

[daon]

For a, you need to show y_n >= y_(n+1) for all n. Hint: For a particular n, a_n is either the supremum or it is not (a_n=y_n or a_n =/= y_n). The fact that y_n is boulded below is shown by using the assumption that a_n is bounded.