Cauchy sequence

**cgiulz** · September 29th 2009, 12:37 PM

Prove that every convergent sequence is a Cauchy sequence.

[SOLVED] I will post my result if anyone is curious.

Thanks

**malaygoel** · October 1st 2009, 02:19 AM

I would like to see your result

**cgiulz** · October 1st 2009, 08:36 AM

Suppose $a_{n}$ $\rightarrow$ $a$ .

$\forall$ $\epsilon > 0$ , $\exists N$ $: |a_{n} - a| < \epsilon/2$ , $\forall$ $k> N$ . Then,

$|a_{n} - a_{m}| = |a_{n} - a + a - a{m}|$

Which is, $< |a_{n} - a| + |a - a_{m}|$

Which is, $< \epsilon/2 + \epsilon/2 = \epsilon$ $\forall$ $n,m > N$ .

I recently discovered how to overuse quantifiers

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