Proof of the following matrix norm inequality$\|A\|_F \leq \sqrt{n} \|A \|_2$
I'm stucked in this one. I tried to rearrange $\| A \|_2=\max_{x \neq o}\frac{\sqrt{\sum_i |\sum_{j} a_{ij}x_j|^2}}{\|x\|_2}$ and writing the Frobenius norm as $\| A\|_F=\frac{\sqrt{(\sum \sum |a_{ij}|^2)\|x\|_{2}^2}}{\|x\|_2}$ but I don't know where to go from there.
We have $$\|A\|_F^2=\sum_{i=1}^n\|Ae_i\|^2\le \sum_{I=1}^n\|A\|_2^2\|e_i\|^2=n\|A\|_2^2$$ where $e_i$ are the elements of the standard basis. Thus the inequality follows.