Help me understand this proof of R^infinity's infinite-dimensionality

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The proof goes as follows:

For contradiction, assume there exists |s|< ∞ such that s = {e1, e2, ... , en} and span(s} = ℝ^∞.

The above makes at least some sense to me. The proof goes on...

Let m > n and u = en+1 + en+2 + ... + em

u \notin span(s), u \in s

Because {e1, e2, ... , en} \notin s, this implies a contradiction. Therefore ℝ^∞ is infinite-dimensional.

I don't see what we just did, and I don't see how what we just did proves that R^infinity is infinite-dimensional. Please help!
 
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In words:
Assume that there is a finite base with a finite number of base vectors e1 to en*.
Consider en+1 (+maybe more vectors) - where it is unclear in which basis that is supposed to be*. This vector cannot be represented by a sum of e1 to en, therefore the initial assumption is wrong and a finite base cannot exist.

* I think the combination of those are a serious problem. While is possible to name (arbitrary) base vectors with ei, this cannot be extended to other vectors. If ei refer to the standard base, you cannot simply assume that a finite base uses those vectors.
 

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