The following theorem states (Monotone Convergence): Let (sn), n=1 to infinity,

Question

The following theorem states (Monotone Convergence):
Let (sn), n=1 to infinity, be a sequence of real numbers. If (sn) is bounded and monotone, then it is convergent.

We are supposed to prove this using three steps, but i'm not so sure how to go about it:
(a)Argue why we may assume WLOG that (sn), n=1 to inifinity, is increasing.
(b)Argue why S=sup{sn|n is an element of N} is finite.
(c)Show that lim sn (as n goes to infinity) = S

Explanation / Answer

assume WLOG that (sn), n=1 to inifinity, is increasing. -- > An assumption to prove why S=sup{sn|n is an element of N} is finite. Since the sequence is convergent the sup is finite lim sn (as n goes to infinity) = S where S is the supremeum of the sequence -- also true

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