Basis for Vector Space of Sequences

I'm trying to understand sequence spaces as vector spaces. I know that various sequence spaces are vector spaces (over $\mathbb{R}$, say) with componentwise addition and scalar multiplication, such as $$\{ (x_n) \mid x_n \in \mathbb{R} \} = \{ \text{ real-valued sequences}\}$$ $$\{ (x_n) \mid x_n \in \mathbb{R},\; \lim x_n =0 \} = \{ \text{ real-valued sequences converging to zero}\}$$ $$\{ (x_n) \mid x_n \in \mathbb{R},\; \lim x_n \text{ exists} \} = \{ \text{convergent real-valued sequences}\}.$$

However, these spaces do not have bases, in the sense that there is no collection of elements such that every sequence can be expressed as a finite linear combination of basis elements. You need infinite combinations of the standard basis vectors $e_i$, (where $(e_i)_n = 1$ if $n =i$, and zero else).

I've been reading through standard proofs of the existence of bases, such as this one, but I can't see where they break down in these cases.

Can anyone point out what goes wrong, or where I may be misunderstanding the definitions? Thanks.

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Those spaces do have bases (any vector space does). These do just not consist simply of the standard basis vectors from the finite dimensional subspaces. –  Tobias Kildetoft Feb 8 at 16:48
And infinite linear combinations are not defined at all. (You are talking about vector spaces, not Banach spaces or alike) –  Martin Brandenburg Feb 8 at 17:37