The Monotone Convergence TheoremDate: 10/02/98 at 15:48:15 From: steven flarity Subject: Monotone convergence theorem Could you please explain the meaning and purpose of the monotone convergence theorem? A previous question stated it as: The Monotone Convergence Theorem states that any nondecreasing or increasing sequence which is bounded above converges. Also, any nonincreasing or decreasing sequence which is bounded below converges. Isn't it obvious that a number sequence that's always increasing, but never goes beyond a certain value, has to converge? How would one ever prove this theorem? Date: 10/03/98 at 01:12:20 From: Doctor Mike Subject: Re: Monotone convergence theorem Hi, This is an interesting and very good question. This is pretty "obvious" and at the same time tricky to prove. The big stumbling block is to find out exactly what it converges to. The sequence: 2.9 , 2.99 , 2.999 , 2.9999 , 2.99999 , .... clearly converges to three. But what would you do if you were told only that it was a monotonic increasing sequence bounded above by six? That is true, but not particularly helpful if you are trying to prove that it converges to something in particular. Some calculus texts do prove this, by using a property of real numbers called completeness, which says that every set of reals with some upper bound, has a least (or smallest) upper bound, often abbreviated to lub. Then it is fairly easy to prove that the sequence converges to that lub. You prove it directly from the definition of convergence of a sequence, you know, with epsilons and all that. I hope this helps. - Doctor Mike, The Math Forum http://mathforum.org/dr.math/ |
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