Suppose (X_n) converges. Show that (X_n) must have a convergent subsequence.?

Answer 1

This follows directly from two facts.
1. All convergent sequences are bounded.
2. The Bolzano-Weierstrass Theorem. All bounded sequences contain convergent subsequences.

Alternatively, you could simply let your subsequence be the sequence itself. ;-)

Answer 2

The result seems rather timid in view of the fact that if the sequence converges, then EXERY subsequence converges to the same limit.