Limit of Absolute Value

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

Then:
 * $\left\vert{x - \xi}\right\vert \to 0$ as $x \to \xi$

where $\left\vert{x - \xi}\right\vert$ denotes the Absolute Value.

Proof
Let $\epsilon > 0$.

Let $\delta = \epsilon$.

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

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

Hence the result.