TITLE: Exponential function and uniform convergence of polynomials.
QUESTION [7 upvotes]: How can I prove that no sequence of polynomials converges uniformly to the exponential function?
Thanks in advance for any help.

REPLY [5 votes]: Suppose that $f:\mathbb R\to\mathbb R$ is a uniform limit of polynomial functions.  Then $f$ is a polynomial function.
Proof Sketch: Suppose $(p_n)$ is a sequence of polynomial functions converging uniformly to $f$.  Then there is a positive integer $N$ such that $m,n\geq N$ implies $|p_n(x)-p_m(x)|<1$ for all $x\in\mathbb R$.  That means that the polynomial function $p_n-p_m$ is bounded, hence constant. Thus our sequence of polynomials has the form $(p_1,p_2,\ldots,p_N,p_N+a_1,p_N+a_2,p_N+a_3,\ldots)$ for some sequence $(a_n)$ of real numbers.  For the limit to exist it must be the case that $(a_n)$ converges to some real number $a$, and $f(x)=p_N(x)+a$ for all $x$.