Proof of Uncountable Basis for $\mathbb{N} \to \mathbb{R}$ over $\mathbb{R}$

Here is a diaginalisation argument.

Let $\{f_i\}$ be a countable set we find $g\not \in span \{f_i\} $.

Construct $g$ as follows.

Look at the vector $$(f_0(0), f_0(1))$$ define $(g(0),g(1))$ not to be a linear multiple of this vector. Now look at the vectors

$$(f_0(2), f_0(3), f_0(4))$$ $$(f_1(2), f_1(3), f_1(4))$$ define $(g(2),g(3),g(4))$ not a linear combination of these vectors. Now look at the vectors $$(f_0(5), f_0(6), f_0(7),f_0(8))$$ $$(f_1(5), f_1(6), f_1(7),f_1(8))$$ $$(f_2(5), f_2(6), f_2(7),f_2(8))$$

define $(g(5),g(6),g(7),g(8))$ not in the span of these three vectors, etc. I think the construction in clear, and $g$ is not in the span. Thus there is no countable basis.

There are uncountably many functions $\mathbb N\to\{0,1\}$. Well-order them all, and then remove each one that is a (finite) linear combination of vectors that come earlier in the well-order. The result is, by contruction, a linearly independent set in your vector space. How large is it?

I claim that each of the removed vectors is not just a linear combination of vectors that remain, but a rational linear combination of such vectors. In each case, when we express the new vector as a linear combination of finitely many already-accepted ones, we have to solve an equation involving a matrix that is infinitely tall but has finite width. But because all elements are either $0$ or $1$, there are actually only finitely many different rows in the matrix, so we can find the coefficients by doing ordinary finite-dimensional linear algebra over $\mathbb Q$.

Now, if the reduced set of $0,1$-vectors were countable, there would be only countably many different finite rational combinations of them -- but this contradicts the fact that there are uncountably many vectors to express.

So we have an uncountable linearly independent set in your vector space, and if we extend that to a basis, we get an uncountable basis.

I don't have a proof with Zorn's Lemma, but probably you like this one aswell:

The space you consider contains $\ell^\infty$, the space of bounded sequences, so if we can show that this space has an uncountable base the same is true for the space of all sequences.

The space $\ell^\infty$ can be made a Banach-space if we consider the supremum-norm on it. But infinite dimensional Banach-spaces can't have a countable Hamel-Base which is due to Baire's theorem.

If $(b_n)_{n\in\mathbb N}$ was a Hamel-base of $\ell^\infty$ then $\ell^\infty = \bigcup_{n\in\mathbb N}\operatorname{span}(b_1,...,b_n)$. Baire's theorem tells us that one of the sets in the union has nonempty interior, thus is all of $\ell^\infty$ because it is also a subspace. This is a contradiction because it would make $\ell^\infty$ finite-dimensional.