In the space of polynomials with degree less than $N+1$: $\operatorname{span}\{ 1,x,\ldots,x^N\}$ defined on $[a,b]$, if a sequence has a uniform convergent limit(under maximum norm in continuous function space).
1) Is the limit a polynomial?
2) Is the limit polynomial's degree less than $N+1$?
3) Do the coeffcients of polynomial sequence converge?