Prove that metric space of polynomial is not complete

Question

Prove that the metric space $P[a,b]$ of all polynomials with uniform metric $d(f,g) = \max|f(t)- g(t)|$ for every $f, g$ belonging to $P[a,b]$ and $t$ belonging to $[a,b]$ is NOT complete.

My approach: I need to find a sequence of polynomials that converges and then show that limit of this sequence does not belong to $P[a,b]$. I can't figure out what can be the suitable polynomial sequence here?

I tried to guess $f_n(t)$ as follows:

$f_n(t)= t$ , for $t$ belonging to $[a,1/n]$ ; and $fn(t)= 1/n$, for $t$ belonging to $[1/n,b]$

Am I going in the right direction? Any suggestions/solutions are welcome!

Thanks

Answer 1

Yes you're going to the right direction. Hint : look at the exponential.

Edit : No, $f_n$ like this is not even polynomial.

Answer 2

$(1+\frac x n)^{n}$ is a polynomial for each $n$ and this converges to $e^{x}$ in the metric $d$ but $e^{x}$ is not a polynomial.