How to prove the non-existence of a polynomial series that uniformly converges to this function? I was asked to prove the following. 

There exists a function $p(x)\in C(a,b)$ $(-\infty<a<b<+\infty)$, such that there does not exists a polynomial  series which uniformly converges to $p(x)$ on $(a,b)$. 

I considered $p(x)=e^{\frac1x}$ on $(0,1)$. If there should exist $p_n(x)\rightrightarrows p(x)$ on $(0,1)$, then let $x_n=\frac1n$, I intended to show that $|p_n(1/n)-e^n|\to\infty$ as $n\to\infty$. But I didn't know how to tackle it. Would LHR help? 

Your example certainly suffices, but I don't think you'll have any luck working with the expression $|p_n(\frac{1}n)-e^n|$ since, for appropriate choice of $p_n$, we can make that $0$ for all $n$ (even if it approximates every other point terribly).
The important thing to note is that $p(x)$ is unbounded, whereas all polynomials are bounded. Therefore, for any $n$, we have:
$$\lim_{x\rightarrow 0}|p_n(x)-p(x)|=\infty$$
which contradicts uniform convergence.
