Limit of Real Function of 2 Variables/Examples/x^2 + y^2 at 0

Example of Limit of Real Function of 2 Variables
Let $f$ be the real function of $2$ variables defined as:
 * $\map f {x, y} = x^2 + y^2$

Then:
 * $\ds \lim_{\substack {x \mathop \to 0 \\ y \mathop \to 0} } \map f {x, y} = 0$

where $\lim$ denotes the limit of $f$.

Proof
Let $\epsilon$ be arbitrary.

Let $\delta = \sqrt {\epsilon / 2}$.

Then we have:
 * $\size x < \sqrt {\epsilon / 2}$

and:
 * $\size y < \sqrt {\epsilon / 2}$

together imply that:
 * $x^2 + y^2 < \epsilon$

Hence the result.