Prove that a sequence converges iff it is a Cauchy sequence.

So the sequence $\displaystyle a_n \to L $. For $\displaystyle \varepsilon > 0 $ there is an $\displaystyle N \in \bold{N} $ such that $\displaystyle j \geq N \implies |a_{n}-L| < \varepsilon/2 $. And so $\displaystyle m,n \geq N \implies |a_{m}-a_{n}| = |a_{m}-L +L-a_{n}| \leq |a_{m}-L|+|L-a_{n}| < \varepsilon $ (by triangle inequality).

Now suppose $\displaystyle a_n $ is a Cauchy sequence. Choose $\displaystyle \varepsilon = 1/2 $. Then there is an $\displaystyle N \in \bold{N} $ such that $\displaystyle n,m \geq N \implies |a_{n}-a_{m}| < 1/2 $. So $\displaystyle a_N - 1/2 < a_n < a_N+1/2 $.

Is this the correct way of proving this?