How unboundedness of $\|T_n\|$ will imply that X is not complete?

Question

I have to show that

The normed space $X$ of all polynomials with norm defined by $$\|x\|=\max\vert\alpha_j\vert$$ ($\alpha_0,\alpha_1,...$the coefficients of $x$) " is not complete

using

Uniform Boundedness theorem:

Let $(T_n)$ be a sequence of bounded linear operators $T_n: X\rightarrow Y$ from a Banach space $X$ into a normed space $Y$ such that ($\|T_nx\|$) is bounded for every $x\in X$, say, $\|T_nx\|\le c_x$ $n=1,2,···$ , where $c_x$ is a real number. Then the sequence of the norms $\|T_n\|$ is bounded, that is, there is a c such that $\|T_n\|\le c$ $n=1,2,··· $.

I don't want the proof of the above claim.I just need to know How unboundedness of $\|T_n\|$(What is the argument/statement/theorem) will imply that X is not complete?

Answer 1

Suppose $X$ is complete. Then $T_n$ satisfy the conditions of the Uniform Boundedness theorem, and hence the sequence of norms $\|T_n\|$ is bounded, a contradiction.

Answer 2

Considerer $P_n(p(x))=na_n$, where $a_n$ is a coefficient of $x^n$ of $p.$