Re-working homework problems that I missed in preparation for exam. I originally lost points on this problem because I applied the product of limits which was incorrect because $\displaystyle{\lim_{x\to 0}(\cos{\frac{1}{\sqrt{x}}})}$ does not exist. Please verify the following proof (I've been struggling with epsilon-delta proofs):
$$ \left| \sqrt{x} \cos{\frac{1}{\sqrt{x}}}\right|<\epsilon\\ \implies\left|\sqrt{x}\cos{\frac{1}{\sqrt{x}}}\right|^2<\epsilon^2\\ \implies\left|x\cos^2{\frac{1}{\sqrt{x}}}\right|<\epsilon^2\\ \implies\left|x\right|<\epsilon^2=\delta~~~~~~\mbox{(because }\cos^2(\frac{1}{\sqrt{x}})\leq 1)\\ \therefore |x|<\delta\implies\left| \sqrt{x} \cos{\frac{1}{\sqrt{x}}}\right|<\epsilon\\ AND\\ \lim_{x\to 0}\left(\sqrt{x}\cos{\frac{1}{\sqrt{x}}}\right)=0 $$