Limit of Absolute Value

Theorem
Let $$x, \xi \in \mathbb{R}$$ be real numbers.

Then $$\left|{x - \xi}\right| \to 0$$ as $$x \to \xi$$.

Proof
Let $$\epsilon > 0$$.

Let $$\delta = \epsilon$$.

From the definition of limit, We need to show that $$\left|{f \left({x}\right) - 0}\right| < \epsilon$$ provided that $$0 < \left|{x - \xi}\right| < \delta$$, where $$f \left({x}\right) = \left|{x - \xi}\right|$$.

Thus, provided $$0 < \left|{x - \xi}\right| < \delta$$, we have:

$$ $$ $$

Hence the result.