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. Proposition 4 Every subsequence of a convergent sequence converges to the same limit. Suppose and b is a subsequence of a , say using . Thus as well.
Every sequence has a monotone subsequence.
The resulting subsequence is monotone increasing. A similar argument produces a monotone decreasing subsequence if the limit is . Note that if a sequence a n is unbounded, then it will have a subsequence which converges to either . Suppose is bounded. Then since there are an infinite number of natural numbers for each n , and thus either or or both.