Every Cauchy Sequence is Bounded Proof

Every Cauchy Sequence is Bounded Proof

[mu-maths]

Tell me if I am right or wrong.
We are giving values to epsilon only to make things look less shabby.
Also, we need a solid bound and that's why we fixed m. Here we took m=N+1 but we could have taken any fixed m which is more than N (or even equal). e.g. m=N+2 or m=N+3.
This is what I interpreted out of this. Please let me know if this is correct interpretation.

[rajat]

Hi. How are we ensuring that in the first N terms of the sequence the max value{max(a1, a2, .., aN)} is finite. Also, how is a(N+1) finite. Everywhere i saw the proof is done like this but no mention of the doubt i am having.

[mrimbord]

this might sound stupid but need help please
see towards the end with M=max{stuff}
how do we know that a1 or a2 or any value between aN are actual numbers and not infinity? like what if M = max{stuff} = infinity?
then all an < infinity but that doesnt mean they're bounded cos everything is less than infinity really. thanks in advance to anyone that answers

[debendragurung]

OK here we have considered ε= 1, for a corresponding N. So in theory we could have taken any other number like ε=2 for a different N∈Ν.