Convergent Sequence in Metric Space has Unique Limit/Proof 1

Proof
Suppose $\displaystyle \lim_{n \to \infty} x_n = l$ and $\displaystyle \lim_{n \to \infty} x_n = m$.

Let $\epsilon > 0$.

Then, provided $n$ is sufficiently large:

So $0 \le \dfrac {d \left({l, m}\right)} 2 < \epsilon$.

This holds for any value of $\epsilon > 0$.

Thus from Real Plus Epsilon it follows that $\dfrac {d \left({l, m}\right)} 2 = 0$, that is, that $l = m$.