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I just wanted to ask whether my proof is correct:

Suppose instead that $\mathbb{R}$ had a countable $\mathbb{Q}$-basis, say $v_1,v_2,v_3,\ldots$ (possibly finite).

Since $\mathbb{Q}$ is countable, $\,\text{span}(v_1,\ldots,v_k)$ is countable for each $k$ (possibly finitely many).

We have $\mathbb{R}=\bigcup_{k}\text{span}(v_1,\ldots,v_k)$ which is a countable union of countable sets.

It follows that $\mathbb{R}$ is countable. Contradiction.

I would be very grateful for any feedback.

Best wishes!

user166724
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1 Answers1

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You admit the possibility that your basis is finite, say of size $k$ elements, but then you go on to discuss the span of the first $n$ elements for any $n \in \mathbb{N}$. If $n > k$, then this doesn't make sense.

Not a huge problem, and your proof looks fine to me otherwise.

syusim
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