Evidently then and moreover . Since was arbitrary the conclusion follows (technically you have to adjust the above argument so that but that's easy enough).
Conversely, suppose that . Then, there exists some such that for every there exists some such that . Evidently this implies that for every one has that . So, let be arbitrary. Then, by definition there exists some such that for every one has that . Consequently
Thus, . Since was arbitrary it follows that .
That should give you the basic idea...I might have skipped a minor detail in the case but it's easily fixable.