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## Main Question or Discussion Point

BTW. theorem 1.1 just states that Every bounded and monotonic sequence is convergent.

- Thread starter shinobi20
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BTW. theorem 1.1 just states that Every bounded and monotonic sequence is convergent.

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lavinia

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What don't you understand about the proof?

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I don't understand the proof of the Bolzano-Weierstrass Theorem according to Serge Lang...

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lavinia

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Yes but what about it don't you understand?

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How can Cn be increasing? How can we be sure that the Xn will less than Xn+1???

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lavinia

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##C_{n}## is the greatest lower bound of the sequence of ##x_{n}##'s except for the first ##n-1## of them. If you remove some more of the ##x_{n} ##'s then the greatest lower bound can not be less than ##C_{n}##.How can Cn be increasing? How can we be sure that the Xn will less than Xn+1???

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- #8

lavinia

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Because they are greatest lower bounds. It doesn't matter if the X's oscillate. ##C_{n}## is lower than all of them except the first ##n-1##. ##C_{n+1}## is lower than all of them except one less so that one removed might be very low.

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