$\displaystyle \lim_{x \to 0} (e^x-x-1) \cos^2 \frac{2}{x} + x + 1$

I know that I need to use the squeeze theorem, and I have an explanation of how to solve the problem, but I don't understand one of the steps, how does:

$\displaystyle 0 \leq \cos^2 \frac{2}{x} \leq 1 \Rightarrow$

$\displaystyle x+1 \leq (e^x-x-1)\cos^2 \frac{2}{x}+x+1 \leq e^x$ ?