I am trying to figure out what I know to be an easy proof but I am having troubles.
Let $f:D \subset \mathbb{R} \rightarrow \mathbb{R}$ and $c$ be a limit point of $D$. If $\lim\limits_{x \to c}\; [f(x)]^2 = 0,$ prove that $\lim\limits_{x \to c}\; f(x) = 0.$
I understand this should be an easy proof but for some reason I am having trouble. One way I thought was assuming that $\lim_{x \to c} f(x)$ exists we must have that it equals zero because of the algebraic limit laws. But I feel that assuming the limit exists is using what I want to show.
Also I have looked at the fact that since $|f^2(x) - 0 | \epsilon$ whenever $|x - c| < \delta$ I know that $f(x)f(x) < \epsilon$ but I don't know how I could break that up more. Any help would be appreciated. Thank you!