Upper bound of coefficients and continuity of differentiation of polynomials of finite order

For some $n\in\mathbb{N^+}$, let $\mathbb{P}_n,\mathbb{P}_{n-1}$ be the metric space of polynomials of order $n$ and $n-1$ respectively, both with the metric: $$\displaystyle \sup_{x\in[0,1]}\{|P(x)-Q(x)|\}$$ I want to find an upper bond $L$, in expression of $\epsilon$, such that $$\forall P(x),Q(x)\in\mathbb{P}_n, d(P(x),Q(x))<\epsilon\;\Longrightarrow \sum_{k=0}^n |p_k-q_k|<L$$ where $p_k$, $q_k$ are coefficients before the $x^k$ term in respectively $P(x)$ and $Q(x)$.

Could someone provide a very simple proof for such an L? (I'm a beginner so I don't know many theories, please don't go too deep).

Further question:

This step comes up when I was trying to prove that the differentiation map

$D:\mathbb{P}_n \to \mathbb{P}_{n-1}$ is continuous using $\epsilon$, $\delta$ proof. Is there an easy way to show this map is continuous apart from using the method above?

Thank you so very much.

-

Identify $\mathbb{P}_n$ with $\mathbb{R}^{n+1}$ through the function that assigns to each polynomial $P(x)=\sum_{k=0}^np_kx^k$ the vector $(p_k)_{k=0}^n$ of its coefficients. Then $$\|P\|_\infty=\sup_{x\in[0,1]}|P(x)|$$ and $$\|P\|_1=\sum_{k=1}^n|p_k|$$ define norms on $\mathbb{P}_n$. Since all norms on a finite dimensional vector space are equivalent, there exists a constant $C>0$ such that $$\frac{1}{C}\,\|P\|_1\le\|P\|_\infty\le C\,\|P\|_1\quad\forall P\in\mathbb{P}_n.$$ You will have to work a little harder to find an explicit value of $C$ (which might depend on $n$.)