Why space of polynomials not complete while $\ell^\infty$ is complete

Question

$\ell^\infty$ is complete under the metric

$$d(a,b)=\sup_{i}|a_i-b_i|$$

, but the following normed space $X$ of polynomial is not complete

$$\| x \|= \max_j |\alpha_j|$$, where for a polynomial of degree $N_x$, we write $x(t)=\sum_{j=0}^\infty \alpha_j t^j$, ($\alpha_j=0$ for $j>N_x$)

I know how to prove the completeness of the first space and incompleteness of the second space.

But it still feels weird, because two spaces are quite similar.

I would appeciate it if someone can give a concrete example of Cauchy Sequence in $X$, that does not converge.

Because my proof is based on uniform boundedness theorem, without explicitly construct non-converging Cauchy sequence.

Answer 1

The point is that polynomials correspond to the set of all sequences that are eventually zero, which is a linear subspace in $\ell^\infty$ that is not closed, and hence not complete. (In fact this is true regardless of which $p$-norm one chooses.)

As an explicit example, consider the polynomials $p_n(t) = \sum_{j=0}^{n} t^j/j^2$, the sequence of which converges to a (formal) power series, and not a polynomial.