Since the sequence is bounded then there exist only one subsequentional limit by bolzano weierstrass theorem. The limit is an interior point by cantor's theorem in this case. Then the limit is bounded. Therefore the set of all subsequebtional limit become bounded. But i don't understand the converse part. Please say am I right in the first part. And help to prove the converse part.
Prove that if the sequence {f(n)} is bounded, then the set of all subsequentional limits of {f(n)} is bounded.Prove conversely.
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real-analysis
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1Bolzano Weierstrauß states that there is at least one converging subsequence in a bounded sequence, not only one. Take $a_N=(-1)^n$ which is bounded and has two subsequential limits. – 2017-02-17
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1Oh, My mistake to understand. Thanks for example. – 2017-02-17
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0Could you add your effort [here](http://math.stackexchange.com/questions/2111200/prove-that-en-ge-fracn1nn/2111211#2111211)? – 2017-03-12