Are Taylor series enough to prove that the polynomials subspace is not closed in L2

Question

Given the open and bounded set $I$, I am considering the Hilbert space consisting of $L^2$(I) (the set of real square-integrable functions, i.e. $f:I\rightarrow \mathbb{R} $ such that $\int f^2<\infty$) and the square norm. I am trying to prove that the subspace of polynomials with real coefficients in I is closed and I am trying to do so by showing that it does not contain a boundary point. Would it be sufficient to show that using, for example, that the exponential function is the limit function of its Taylor series?