Showing that a set is not closed

Question

Let $C$ be the set of continuous real valued functions on $[0,1]$ with metric

$$d(f,g) = \text{sup} \{ |f(x) - g(x)| \ : x \in \ [0,1]\}$$

Show that the set of Polynomials in $C$ is not closed.

I don't know how to approach this problem.

Please help.
Thanks in advance.

Answer 1

Take the sequence of polynomials given by $$ 1\\ 1+x\\ 1+x+\frac12x^2\\ 1+x+\frac12x^2+\frac16x^3\\ \vdots\\ 1+x+\frac12x^2+\cdots+\frac1{n!}x^n\\ \vdots $$ Does this sequence converge to an element of $C$? Does it converge to a polynomial?