Show that if then the sequence { } converges, but not conversely.
Notice that , so that converges if, and only if is a convergent series.
If a series is absolutely convergent, then it converges. However the converse is not true. Using a counterexample of this property (like , for instance), you can deduce a sequence which converges, but such that diverges.
Let
then we know that
provided that is a nonnegative sequence of reals.
From your question, we see that
holds, i.e. there exists and such that for all , this implies that is a Cauchy sequence (see the definition of Cauchy sequences and let ), and Cauch sequences are always convergent in complete spaces ( is complete).
For the converse part, it suffices to give a counter example.
Let
Then, we see that
And therefore, we get
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On the other hand, we have .
This completes the proof.