Limit of Real Function/Examples/x times Sine of Reciprocal of x at 0/Proof

Proof
By definition of the limit of a real function:
 * $\ds \lim_{x \mathop \to 0} \map f x = A$


 * $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \R: 0 < \size x < \delta \implies \size {\map f x - A} < \epsilon$
 * $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \R: 0 < \size x < \delta \implies \size {\map f x - A} < \epsilon$

Let $\epsilon \in \R_{>0}$ be chosen arbitrarily.

Let $\delta = \epsilon$.

Then we have:

Hence the result.