3.1. Sequences
Examples 3.1.8(c):
Both statements are false. As a counter-example to the first statement, consider the sequence{ (-1) j }Each term of this sequence is bounded by -1 or +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 a finite number.
It does, however, get closer and closer to infinity, but we do not, at this time, consider this convergent.