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.
MathOverflow is a question and answer site for professional mathematicians. It only takes a minute to sign up.
Sign up to join this communityI 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.
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 $P\equiv P_0$.