Closed subspace of vector space

Question

I would like to know where the problem is in the following proof that every finite-dimensional subspace $F$ of a normed vector space $E$ is closed.

Let $(u_n)\in F^{\mathbb{N}}$ be a sequence converging to $l$ and $(e^1,\ldots,e^p)$ be a basis of $F$.

We have, for all $n$, $$ u_n = x_n^1e^1+\ldots x_n^p e^p $$ and it converges to $$ x_\star^1e^1+\ldots x_\star^p e^p =l $$ which, obviously, belongs to $F$.

Answer 1

Your proof is not correct, since it assumes that the limit of your sequence must be of the form$$x_\star^1e^1+\cdots+x_\star^ne^n,$$which is exactly what you are supposed to prove.