Let $f:[a,b]\to \Bbb C$ be a continuous function on $[a,b]$. Then there exists a sequence $\{p_n\}$ of polynomials such that $$p_n\to f \text{ uniformly on }[a,b] \;\;\;(1)$$ Equivalently, for all $\varepsilon>0 $ there exists a polynomial $P$ such that $$\|P-f\|_{\infty}<\varepsilon \;\;\; (2)$$
I want to show that two conclusions are equivalent.
Suppose there exists a sequence $\{p_n\}$ of polynomials such that $$p_n\to f \text{ uniformly on }[a,b]$$ Then for all $\varepsilon>0\;\; \exists n\in \Bbb N$ such that $\forall n>N$,$|p_n(x)-f(x)|< \varepsilon$ for all $x\in[a,b]$. How to choose polynomial $P$ in $(2)$ that can satisfies the second conclusion?
Conversely, If $(2)$ holds, can I just choose $p_n=P\;\; \forall n\in \Bbb N$, so that $(1)$ hold?