Let's define $B_s$ as the of real valued sequences $(x_n)$, such that $sup_{N\in \mathbb N} |\sum_{k=0}^{N}{x_k}| $ is bounded, and make it a vector space considering the usual pointwise operations on sequences and scalars. Then define a norm by $||(x_n)||=sup_{N\in \mathbb N} |\sum_{k=0}^{N}{x_k}| $.
Clearly this space contains properly the set $C_s$ that consist of convergent series( for example $x_n = (-1)^n$).
I want to prove that:
i) $B_s$ is a banach space
$ii)$ $C_s$ is a closed subspace of $B_s$.
I tried by finding explicitly the limit of a Cauchy sequence, I think that if I have a Cauchy Sequence $(x_n)_n$ in $B_s$ given by $x_n=(x_{nk})_k$ , then the limit it's given by the pointwise limit ( I actually proved that the pointwise limit exist) But I can't prove convergence. Please I really need help with this problem )=
It's probably easier if you consider the space $\ell^\infty$ of all bounded (real) sequences and its subspace $c$ of convergent sequences. On $\ell^\infty$, you take the norm $\lVert y\rVert_\infty = \sup \lvert y_k\rvert$.
Then $I \colon \ell^\infty \to B_s$,
$$I(y) = (y_0, y_1 - y_0, y_2 - y_2, \dotsc)$$
is an isometric isomorphism.