solution

This is the solution for problem #1 in my previous post.

Let , since is a Cauchy sequence there exist an such that if if Similarly since a subsequence converges to we have that when . Now let be the larger of and .

Now .

Here is the solution for #2 in my previous post (this one took a little longer for me to think of).

Let be a sequence in such that . Now from properties of we have that for all Now let , and since we have that for large enough. So since is complete. Additionally, is a limit point of each , so .