Let S ≠ 0. Is it true that if |Sn| --> |S|, then Sn --> S?
I think it's true. This is what I have, correct me if I'm wrong...
First suppose that |Sn| --> S. Let e > 0 such that |S - Sn| < e. Then -e < |S - Sn| < e. So -e < ±(S - Sn) < e. which implies that -e < S - Sn < e. So Sn --> S.
I'm not sure if this is the way to do it.
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It's false. For a counterexample, consider Sn = (-1)^n and S = 1 ≠ 0.
Then observe that |Sn| = |(-1)^n| = 1 --> 1 = |S|.
But observe that Sn does not approach any limit, since it oscillates between 1 and -1.
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Note however that the converse is true: "If Sn --> S, then |Sn| --> |S|."