I know that if $P\_n$ are continuous functions and $P\_n \rightrightarrows P$, $P$ is also continuous function. But I can't see in which direction I should dig to prove that $P$ is polynomial.


I will appreciate any hint and help.



If $P_n \rightrightarrows P$ in $\mathbb{R}$ and $P_n$ are polynomials proof that $P$ is polynomial
We have $(P\_n-P\_m)\rightrightarrows 0$ if $n,m$ tend to infinity. Since $P\_n-P\_m$ is a polynomial, this yields that degrees of $P\_n$ are uniformly bounded, say they do not exceed $d$. Now even the pointwise convergence in $d+1$ points yields the coefficientwise convergence (by Lagrange interpolation, for example), hence on any segment $P\_n\rightrightarrows P\_0$ where $P\_0$ is this limit polynomial. Hence $P\equiv P\_0$.


