Example: Is it true that a bounded sequence converges ? How about monotone increasing sequences ?
Both statements are false. As a counter example to the first statement, consider the sequence Each term of this sequence is bounded by -1 and +1, so that the sequence is indeed bounded. But, as we have seen before, the sequence does not converge.

As for the second statement, consider the simple sequence {n}, i.e. the sequence consisting of the numbers {1, 2, 3, 4, ...}. It is obviously increasing, but does not converge to any finite number.

It does, however, get closer and closer to infinity, but we do not, at this moment, consider this convergent.

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