I have a question about lecture notes on Donsker's theorem Davar Khoshnevisan from this page https://www.math.utah.edu/~davar/ps-pdf-files/donsker.pdf
On the page 4 there is a equality: $ \sup_{t \in [0,1]} \mathscr{S}_n(t) = \max_{0 \leq j \leq n} \frac{S_j}{\sqrt{n}} $ with comment "by convexity".
Firstly does any proof is needed here? And secondly at the beginning of page 7 exactly this relation is used?
Let's make a few observations:
(i) $\mathscr{S}_n(t)$ is continuous in $t$
(ii) For any integer $j\in[0,n]$, we have $\mathscr{S}_n\left(\frac jn\right)=\frac{S_j}{\sqrt n}$
(iii) For $t\in\left(\frac jn,\frac{j+1}n\right]$, the function $\mathscr{S}_n(t)$ is linear in $t$, therefore $\,\sup_{t\in\left(\frac jn,\frac{j+1}n\right]}\mathscr{S}_n(t)\,$ is achieved at either $t=\frac jn$ or $\ t=\frac{j+1}n$, hence $$\sup_{t\in\left(\frac jn,\frac{j+1}n\right]}\mathscr{S}_n(t)=\max\left(\frac{S_j}{\sqrt n},\frac{S_{j+1}}{\sqrt n}\right).$$
Then one can conclude that $$\sup_{t\in[0,1]}\mathscr{S}_n(t)=\max_{0\leq j\leq n}\frac{S_j}{\sqrt n}.$$