I need to show that a complete normed space X has no countable Hamel basis.
One possibility is to with Baire's theorem.
I, however, try to give an explicit sequence, namely:
For a contradition, let $\mathcal{B} = \{b_i, i\in\mathbb N \}$ the countable Hamelbasis of X.
W.l.o.g. $B_i\ne 0$ for all $i$.
Define: $x_n=\sum_{i=1}^n \frac{1}{i^2} \frac{b_i}{\|b_i\|}$. Obviously $(x_n)_n$ is Cauchy.
But then it has a limit $x=\sum_{i=1}^\infty \frac{1}{i^2} \frac{b_i}{\|b_i\|}$, which has no finite decomposition $x=\sum_{k=1}^M \lambda_k \frac{b_{i_k}}{\|b_{i_k}\|}$ because the base elements are linearly independed.
Is that also a correct proof?
Thanks.