Identity Mapping on Real Vector Space from Chebyshev to Euclidean Metric is Continuous

Theorem
Let $\R^n$ be an $n$-dimensional real vector space.

Let $d_2$ be the Euclidean metric on $\R^n$.

Let $d_\infty$ be the Chebyshev distance on $\R^n$.

Let $I: \R^n \to \R^n$ be the identity mapping from $\R^n$ to itself.

Then the mapping:
 * $I: \struct {\R^n, d_\infty} \to \struct {\R^n, d_2}$

is $\tuple {d_\infty, d_2}$-continuous.

Proof
Let $a = \tuple {a_1, a_2, \ldots, a_n} \in \R^n$.

Let $\epsilon \in \R_{>0}$.

Let $\delta = \dfrac {\epsilon} {\sqrt n}$.

Let $x = \tuple {x_1, x_2, \ldots, x_n}$ be such that $\map {d_\infty} {x, a} < \delta$.

That is:
 * $\ds \max_{i \mathop \le i \mathop \le n} \set {\size {a_i - x_i} } < \delta$

Then:

The result follows by definition of $\tuple {d_\infty, d_2}$-continuity.