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 $$ p_k\left(x\right)=\underset{i=1}{\overset{l_k}{\sum <!--\; \sum \; -->}}{c}_i{x}^i$$ and therefore

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

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

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

Since $$ L$$ is closed then $$ sup\left(L\right)\in <!--\; \in \; -->L$$, if it's the case that $$ R=\mathrm{\varnothing <!--\; \varnothing \; -->}$$ then we linearly interpolate from $$ f(sup\left(L\right))$$ to $$ 0$$ over $$ [sup\left(L\right),N]$$, otherwise $$ R\ne <!--\; \ne \; -->\mathrm{\varnothing <!--\; \varnothing \; -->}$$ and we linearly interpolate from $$ f(sup\left(L\right))$$ to $$ f(inf\left(R\right))$$ over $$ x\in <!--\; \in \; -->[sup\left(L\right),inf\left(R\right)]$$. 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 $$ f$$.