The Monotone Convergence Theorem

Date: 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/