I am given that the definition of the Schatten $p$-norm is $||A||_p = \left[\text{Tr}(A^{\dagger}A)^{p/2}\right]^{1/p}$
Next we have, $\lim_{p\rightarrow\infty}||A||_p = \max\{||Au||: ||u||=1\}$
1) I'm not able to see this limit and
2) How is $||..||$ in $||Au||, ||u||$ defined?
First for a vector such as $$v=\begin{bmatrix}v_1\\v_2\\\vdots\\v_n\end{bmatrix}$$we define $||v||_p$ as $$||v||_p=\sqrt[p]{\sum_{i=1}^n |v_i|^p}$$also $$\lim_{p\to \infty}||A||_p{=\lim_{p\to \infty}(\sigma_1^p+\cdots +\sigma_n^p)^{1\over p}\\=\lim_{p\to \infty}\sigma_{\max}\left[\left({\sigma_1\over \sigma_{\max}}\right)^p+\cdots +\left({\sigma_n\over \sigma_{\max}}\right)^p\right]^{1\over p}\\=\sigma_{\max}\lim_{p\to \infty} k^{1\over p}\\=\sigma_{\max}}$$where $k$ is the multiplicity of $\sigma_{\max}$ among all the singular values. Also by a simple proof using SVD we can show that $$\max_{||u||=1}||Au||=\sigma_{\max}$$hence the result is obtained.