By considering the function $\mathbb{R}^{n+1} \rightarrow \mathbb{R}$ defined by

$$R\left(a_{0}, \ldots, a_{n}\right)=\sup _{t \in[-1,1]}\left|\sum_{j=0}^{n} a_{j} t^{j}\right|$$

or otherwise, show that there exist $K_{n}>0$ and $\delta_{n}>0$ such that

$$K_{n} \sum_{j=0}^{n}\left|a_{j}\right| \geqslant \sup _{t \in[-1,1]}\left|\sum_{j=0}^{n} a_{j} t^{j}\right| \geqslant \delta_{n} \sum_{j=0}^{n}\left|a_{j}\right|$$

for all $a_{j} \in \mathbb{R}, 0 \leqslant j \leqslant n$.

Show, quoting carefully any theorems you use, that we must have $\delta_{n} \rightarrow 0$ as $n \rightarrow \infty$.