thanks. what does b_n>=a_n mean, the reason i cant answer it is because i dont know what this means
It's a bit surprising you're asked to prove that statement without being told what it means. Then again, you could just blindly apply it in the definition and end up with the proof.
It just means that if we denote
{an}={a1,a2,a3,…}
and
{bn}={b1,b2,b3,…}
and so bn≥an means b1≥a1 and b2≥a2 and b3≥a3 and so on...
So every term of the sequence {bn} is greater than or equal to the respective element of the sequence {an}
so all the terms in b_n are greater than or equal to those of a_n, so if
is it that A sequence (an) tends to infinity if, for every C > 0, there exists
N ∈ N such that a_n > C whenever n > N so b_n tends to infinity?
and as b_n>a_n>C for all n in N b_n tends to infinity
I would not be happy with what you've written, although I suspect you've got the right idea.
The flipside of "this proof has only one line of actual reasoning" is that if you don't give that line of reasoning in a clear way, it's hard to give much credit for what you have done.
Your proof/explanation should look like:
"By definition of an→∞, {definition goes here}".
But we know that bn≥an
Therefore {actual reasoning step}, and so {something looking very like your first definition}
I would not be happy with what you've written, although I suspect you've got the right idea.
The flipside of "this proof has only one line of actual reasoning" is that if you don't give that line of reasoning in a clear way, it's hard to give much credit for what you have done.
Your proof/explanation should look like:
"By definition of an→∞, {definition goes here}".
But we know that bn≥an
Therefore {actual reasoning step}, and so {something looking very like your first definition}