Monotonic bounded sequence theorem

[pakmingki]

So the theorem states if a sequence is monotonic and bounded, it converges.
WEll, it's easy enough to prove is a sequence is monotonic, but how would one go about proving that a sequence is bounded?

 

[HallsofIvy]

?? Are you under the impression that there is ONE way to prove something? How to prove a sequence is bounded depends a lot upon the sequence! For one thing, how do you know it is bounded? Does it have an upper bound or a lower bound? If you feel sure that a sequence has an upper bound, can you guess an upper bound and then try to prove that is correct. Often with sequences, the best way to prove anything is by induction-show that if a_n< b then a_{n+1[/b]< b. Of course, how you would do that would depend on how the sequence is defined.}

 

[radou]

If you're trying to prove the theorem in general, you simply have to write down what it means for a sequence to be bounded and monotonic, and use the supremum axiom, and poke into the definition of the supremum.