How to show that the space of polynomials is not complete

Question

Denote by $P[0,1]$ the set of all polynomials $p\colon [0,1]\to\mathbb{R}$; this is a vector space. Endow $P[0,1]$ with the norm $$\| p\|=\sup_{t\in [0,1]}{| p(t)|}.$$ I want to show that this particular norm on $P[0,1]$ isn't complete.

I think that, for example, the sequence $$p_n(x)=1+x+\dfrac{x^2}{2!}+\cdots+\dfrac{x^n}{n!}\qquad (n\in \mathbb{N})$$ is uniformly convergent on $[0,1]$ to the function $e^{x}$ that isn't a polynomial. Then $(p_n)$ is a Cauchy sequence, but it doesn't converge in $P[0,1]$. Now, how I can prove this, but more formally?

Greetings...

Answer 1

$\left\Vert p_{n}-p_{m}\right\Vert =\sup_{\left[ 0,1\right] }\left\vert %TCIMACRO{\dsum \limits_{i=0}^{n}}% %BeginExpansion {\displaystyle\sum\limits_{i=0}^{n}} %EndExpansion \frac{x^{i}}{i!}-% %TCIMACRO{\dsum \limits_{i=0}^{m}}% %BeginExpansion {\displaystyle\sum\limits_{i=0}^{m}} %EndExpansion \frac{x^{i}}{i!}\right\vert =\sup_{\left[ 0,1\right] }\left\vert %TCIMACRO{\dsum \limits_{i=m+1}^{n}}% %BeginExpansion {\displaystyle\sum\limits_{i=m+1}^{n}} %EndExpansion \frac{x^{i}}{i!}\right\vert \leq% %TCIMACRO{\dsum \limits_{i=m+1}^{n}}% %BeginExpansion {\displaystyle\sum\limits_{i=m+1}^{n}} %EndExpansion \frac{1}{i!}.$ Noting that $% %TCIMACRO{\dsum \limits_{i=0}^{n}}% %BeginExpansion {\displaystyle\sum\limits_{i=0}^{n}} %EndExpansion \frac{1}{i!}\rightarrow e$, so $\left( p_{n}\right) $ is a Cauchy sequence. Now, assume that $p_{n}\rightarrow p\in P\left[ 0,1\right] $. Hence $p_{n}\left( x\right) \rightarrow p\left( x\right) $ $\forall x\in\left[ 0,1\right] .$ So $p\left( x\right) =e^{x}\in P\left[ 0,1\right] .$ Assume $e^{x}=p\left( x\right) =% %TCIMACRO{\dsum \limits_{i=0}^{M}}% %BeginExpansion {\displaystyle\sum\limits_{i=0}^{M}} %EndExpansion a_{i}x^{i}$. Then $e^{x}=p^{\left( M+1\right) }\left( x\right) =0$ which is impossible.