🏗️ $\Theta \rho ϵ\eta \Pi \alpha \tau \pi$🚧 (under construction)

By wat, we obtain a sequence of poly's $\left(p_k\right)$ that converge uniformly o $f$.

We know that for any $k$ we see that $p_k\left(x\right)={\sum }_{i=1}^{l_k}{c}_i{x}^i$ and therefore

Since products maintain uniform convergence, we also know that $f\left(x\right)p_n\left(x\right)\mathrm{\backslash stackrel}unif\to f{\left(x\right)}^2$. And define $H_n:[0,1]\to \mathbb{R}$ as $${\displaystyle H_n\left(x\right):={\int }_0^{x}f\left(t\right)p_n\left(t\right)dt}$$ therefore $H_n\mathrm{\backslash stackrel}unif\to H$ where $${\displaystyle H\left(x\right)={\int }_c^x{f}^2\left(x\right)dx}$$ specifically when $x=1$ we have that $${\displaystyle {\int }_0^{1}f\left(x\right)p_n\left(x\right)dx(=0)\mathrm{\backslash stackrel}unif\to {\int }_0^1{f}^2\left(x\right)dx}$$ thus ${\int }_0^1{f}^2\left(x\right)dx=0$ so that $f=0$ as needed.

We construct the new function $g$ as follows, given $p\in X$ then we define $g\left(p\right)=f\left(p\right)$, otherwise $p\notin X$, note that since $X$ is compact it's closed so that $X^C$ is open, since $p\in X^C$ then we know that there exists some $ϵ\in {\mathbb{R}}^{+}$ such that $B(x,ϵ)\subseteq X^C$

Now define $L=\{x\in X:x<p\}$ and $R=\{x\in X:x>p\}$. If both $L=R=\varnothing$ then $X=\left\{p\right\}$ and so since we assumed $p\notin X$ that would be a contradiction, so instead one of these must be non-empty, assume that $L\ne \varnothing$, we claim that $L=X\cap (-\infty ,p-ϵ]$, this is the case because for any element $s\in B(p,ϵ)$ it's true that $s\notin L$, but also we do know that $p-ϵ\in X$, therefore $L$ is the intersection of two closed sets and therefore closed.

Since $L$ is closed then $sup\left(L\right)\in L$, if it's the case that $R=\varnothing$ then we linearly interpolate from $f(sup\left(L\right))$ to $0$ over $[sup\left(L\right),N]$, otherwise $R\ne \varnothing$ and we linearly interpolate from $f(sup\left(L\right))$ to $f(inf\left(R\right))$ over $x\in [sup\left(L\right),inf\left(R\right)]$. Note that by doing this it maintains the sup norm.

The function is continuous as it's a piecewise continuous and at each connection point they are continouous as the left and right limits are equal. Thus $g$ is a continuous extension of $f$ on $[-N,N]$, due to this we can apply WAT to $g$ to obtain a sequence of polynomials whose limit is $g$, then by restriction the domain of the polynomails to $X$ then these "new" ones go to $f$.