3.1. Sequences

Examples 3.1.8(c):

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
{ (-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.

Next | Previous | Glossary | Map