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Every bded monotone sequence in R is convergent.

The proof:

suppose {x_k} is a bded increasing sequence. Let M be the sup of the set of values {x_1, x_2,....} I claim that x_k -> M.

Since M is an upper bd, we have x_k <= M for all k. (***)

on the other hand, since M is the least upper bd, for any epislon >0, there is some K for which x_K > M - epislon. Since the x_k's increase with k, we also have x_k > M - epislon for all k > K.

therefore M - epislon < x_k <= M for all k > K, and this shows that x_k -> M

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my question is here, I understand most of the proof, but what I want to ask is , why dont we just stop at the line ending with (***) sign above, if we just show M is the least upper bd and x_k is increasing, then we are done, why do we need to show M-epislon < x_k as well?

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# Monotone Sequence Theorem

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