TITLE: Show that $f$ is a polynomial if it's the uniform limit of polynomais
QUESTION [5 upvotes]: Let $f:\Bbb R\to \Bbb R$ be a function which is the uniform limit of polynomials. I want to show that $f$ is a polynomial. 
I mean this seems a bit trivial... 
If it's the uniform limit of the set of polynomials doesn't that guarantee it's a polynomial? 
If I define $f$ to be the set of all polynomial do I define a uniform limit $L$ s.t $d(f,L)<\epsilon$?  or approach via contradiction? 
any help would be great

REPLY [7 votes]: Hint: if $f$ is a non-constant polynomial, then $f$ is unbounded.  What does this say about $f_n(z) - f_m(z)$, if $f_n$ is a sequence of polynomials that converges uniformly on $\mathbb R$?