Convergent Sequence Minus Limit

Theorem
Let $$X$$ be one of the standard number fields $$\Q, \R, \C$$.

Let $$\left \langle {x_n} \right \rangle$$ be a sequence in $X$ 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 $$\lim_{n \to \infty} x_n = l$$.

Alternative Proof
We note that all of $$\Q, \R, \C$$ can be considered as metric spaces.

Then under the usual metric, $$d \left({x_n, l}\right) = \left|{x_n - l}\right|$$.

The result follows from the definition of metric: $$d \left({x_n, l}\right) = 0 \iff x_n = l$$.