Homework: Real analysis limit of Matrix function Can it be proven that 
$$
\lim_{A \to 0}{\frac{A^T A}{||A||}} = 0
$$
where $||\cdot||$ is the operator norm and $A \in M_{m \times n}$? 
I think that $\frac{A}{||A||}$ should be constant, and that $||A^T|| = ||A||$, so $\lim_{A \to 0}{A^T}$ should be zero, but I feel that I am missing something about how to write it up...

As $\|\cdot\|$ is the operator norm, we have $\|A^TA\|=\|A\|^2$. So
$$
\left\|\frac{A^TA}{\|A\|}-0\right\|=\frac{\|A^TA\|}{\|A\|}=\frac{\|A\|^2}{\|A\|}=\|A\|\to0
$$
