Show that if a sequence has a bounded subsequence , it has a convergent a convergent susequence
Now, I presume you know that any non-decreasing sequence of real numbers, having an upper bound, converges (and, of course, any non-increasing sequence of real numbers, having a lower bound, converges). That's a defining property of the real numbers.
What you need to do is show that every sequence contains a monotone subsequence. Here's a hint: Let S be the set of indices, i, such that if j> i, then . Consider two cases: S is infinite or S is either empty or finite.