Intermediate steps to this simplifcation

Question

Here's the problem

And immediately after is the solution, with no intermediate steps

What steps occurred in order to obtain the inequality after $Note$?

Answer 1

It is known that for all $x$, $\sin (\frac{1}{x})^2 \le 1$

This implies that $-1\le \sin(\frac{1}{x}) \le 1$ was used.

In general,$-|t| \le t\sin(\frac{1}{x}) \le |t|$.

In this case, $t=\sqrt [ 5 ]{x } $