Convergent Sequence Minus Limit

Theorem
Let $$\left \langle {x_n} \right \rangle$$ be a sequence in $\mathbb{R}$ which converges to $$l$$.

That is, $$\lim_{n \to \infty} x_n = l$$.

Then $$\lim_{n \to \infty} \left|{x_n - l}\right| = 0$$.

Proof
Let $$\epsilon > 0$$.

We need to show that there exists $$N$$ such that $$\forall n > N: \left|{\left({\left|{x_n - l}\right| - 0}\right)}\right| < \epsilon$$.

But $$\left|{\left({\left|{x_n - l}\right| - 0}\right)}\right| = \left|{x_n - l}\right|$$.

So what needs to be proved is just $$x_n \to l$$ as $$n \to \infty$$, which is the definition of $$\lim_{n \to \infty} x_n = l$$.