unsure of a proof

**worc3247** · January 3rd 2011, 01:57 PM

Ok so the question is:
Prove carefully that a convergent sequence is bounded.
Here is my proof thus far, but i'm not really sure if it is correct\finished:
$|a_n| = |a_n - L + L| \leq |a_n - L| + |L|$

$a_n$ is convergent, say to a limit L so:
Let $\epsilon > 0$
$\exists N \in \mathbb{N}$ such that $\forall n \geq N |a_n - L| < \epsilon$ .

$|a_n| \leq |a_n - L| + |L| < \epsilon + |L|$ $\forall n \geq N$ .

Is this enough to show that it is bounded?

**Also sprach Zarathustra** · January 3rd 2011, 02:00 PM

Yes, but I will wrote a better proof...

$\{a_n\}^{\infty}_{n=1}$ convergent say to $L$ hence by definition:

Exist $N\in\mathbb{N}$ so for all $n>N$ : $|a_n-L|<1$ .

Let us mark $M=max\{|a_1|,|a_2|,...,|a_N|,|L|+1\}$ and then if $n\leq N$ then we have: $|a_n|\lwq M$

And if $n>N$ then:

$|a_n|\leq |L|+|a_n-L|<|L|+1\leq M$

Hence $M$ is upper bound to sequence $\{|a_n|\}^{\infty}_{n=1}$ , the conclusion is that $\{a_n\}^{\infty}_{n=1}$ bounded.

This is more elegant proof.