Countable basis of function spaces

Show that the space of functions $f: \mathbb{N} \to \mathbb{R}$ does not have a countable basis.

I really don’t know where to start with this one! Could anyone help me?


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Try this. Since $\mathbb N$ and $\mathbb Q$ have the same cardinal, it is the same question to ask whether the set of functions $f : \mathbb Q \to \mathbb R$ has a countable basis. For $x \in \mathbb R$, define $g_x : \mathbb Q \to \mathbb R$, by
g_x(s)=1\quad\text{if }s<x,\qquad g_x(s)=0\quad\text{if }s\ge x
for all $s \in \mathbb Q$. Then the set $\{g_x : x \in \mathbb R\}$ is linearly independent and uncountable.