# Monotone convergence theory

1. Jul 21, 2004

### jmazurek

Is the statement "Every bounded and monotone sequence of real numbers is convergent" proved my the monotone convergence theory? Thanks!!

2. Jul 21, 2004

### AKG

3. Jul 22, 2004

### HallsofIvy

?? That IS the "monotone convergence theorem".

One way to prove it is to define the real numbers as "equivalence classes of monotone sequences" where {an} is equivalent to {bn} if and only if {an-bn} converges to 0.

Another way is to define real numbers as Dedekind cuts which makes it easy to prove the "least upper bound" property and use that to prove the least upper bound theorem.

The "least upper bound property", "monotone convergenence", "Cauchy Criterion", "connectedness of the real numbers", and "every closed and bounded set is compact" are all equivalent- given any one you can prove the others. They are all "fundamental" in the sense that you can define the real numbers in ways that make it easy prove on or the other of these.