I know that every Cauchy sequence is bounded, but is the reverse true?
Asked
Active
Viewed 7,923 times
7
-
Whether every bounded sequence has Cauchy subsequence is a bit more interesting question. – Serge Seredenko Nov 25 '16 at 18:04
1 Answers
24
No. Consider the sequence $$1,-1,1,-1,1,-1,\dots$$ Clearly this seqeunce is bounded but it is not Cauchy. You can show this directly from the definition of Cauchy.
Alternatively, every Cauchy sequence (in $\mathbb{R}$) is convergent. Clearly the above sequence is not, thus it is not Cauchy.
Mathematician 42
- 12,591