Help with Spivak's treatment of epsilon-N sequence definition

[mitcho]

I have just started my first real analysis course and we are using Spivak's Calculus. We have just started rigorous epsilon-N proofs of sequence convergence. I was trying to do some exercises from the textbook (chapter 22) but there doesn't seem to be any mention of epsilon-N in the solutions (apart from the first one). Are they still rigorous proofs or what? Can you rigorously prove that a sequence converges without appealing to the definition?

It would also be good if someone who had the text could take a quick look at the solutions to the exercises at the end of chapter 22 (third edition) and see what they think about them.

Thanks

 

[wisvuze]

you don't always have to appeal directly to the definitions. For example, if you have a non-negative sequence of numbers { a_n }, and if you furthermore know that the non negative sequence of numbers { b _ n } converges to zero* with a_ n < b _ n , then it is clear that {a_n } converges ( of course, the proof of this theorem would have required some delta-epsilon proof ).

Or, if you have an increasing sequence that is bounded, then it converges.