Nth Root of 1 plus x not greater than 1 plus x over n

Theorem
Let $x \in \R_{>0}$ be a (strictly) positive real number.

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Then:
 * $\sqrt [n] {1 + x} \le 1 + \dfrac x n$

Proof
From Bernoulli's Inequality:
 * $\left({1 + y}\right)^n \ge 1 + n y$

which holds for:
 * $y \in \R$ where $y > -1$
 * $n \in \Z_{\ge 0}$

Thus it holds for $y \in \R_{> 0}$ and $n \in \Z_{> 0}$.

So: