This section gives some consequences of a sequence having a limit. These results are useful sometimes to show that a sequence does not converge.
Every convergent sequence is bounded.
Suppose that is a convergent sequence. The sequence is then bounded above by the largest element of and bounded below by its smallest element. Because of this proposition we know that if a sequence is not bounded ten it does not converge. We can also show that a sequence does not converge by demonstrating two subsequences which tend to different values. A subsequence is thus a sequence with some of the terms left out.