Question

Let \[0 \le x \le 1\] \[\epsilon > 0\] and n be a positive integer.

Prove that

\[(x + \epsilon)^\frac {1}{n} \le x^\frac{1}{n} + \epsilon^\frac{1}{n}\]

Answer 1

One can see analytically that they can only be equal if they're the same, however if they aren't then the latter will always be greater.