# Thread: How to prove that all cauchy sequences are bounded?

1. ## How to prove that all cauchy sequences are bounded?

Hi I was just wondering if anyone could help me out a little with this question?
Thanks

2. Since $\displaystyle \{a_n\}$ is Cauchy, there exist N such that if m,n are both larger than N, then $\displaystyle |a_n- a_m|< 1$. Let M be the largest of $\displaystyle |a_n- a_m|$ for m and n both less than or equal to N+1.

Now, show that $\displaystyle |a_n- a_m|< M+1$ for all n and m.
(Consider the cases: (a) n and m both less than N, (b) n and m both larger than N, (c) one of m or n less than N and the other larger.)

Finally show that for all n, |a_n|< |a_1|+ M+1.

3. Originally Posted by binkypoo
Hi I was just wondering if anyone could help me out a little with this question?
Here is a neat trick using what Halls has done.
$\displaystyle n \geqslant N\, \Rightarrow \,\left| {x_n - x_N } \right| < 1\, \Rightarrow \,\left| {x_n } \right| < 1 + \left| {x_N } \right|$

So $\displaystyle \left( {\forall k} \right)\left[ {\left| {x_k } \right| < 1 + \sum\limits_{j = 1}^N {\left| {x_j } \right|} } \right]$