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I'm reading a book on analysis independently. There is a Theorem thatRis complete, i.e. any Cauchy sequence of real numbers converges to a real number.

He lets a1, a2, a3, ... be a Cauchy sequence, then considers the the set:

S = { x an element ofR: x ≤ a_{n}for an infinite number of positive integersn}

and the proof shows that lim a_{n}= supS.

I'm baffled at what the set S is supposed to be. The proof won't work if it is the intersection of sets { x : x ≤ a_{n}} for all n, nor union of such sets. It can't be the limit of a_{n}because this is a proof of it's existence.

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# Homework Help: R is complete?

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