- #1
happyg1
- 308
- 0
Hi,
Here is the question:
Prove that if the sequence {s} has no convergent subsequence then {|s|} diverges to infinity.
To me, this seems so easy, but I'm having a really hard time putting it down in a rigorous manner.
My thoughts are:
every convergent sequence has a convergent subsequence (theorem in the book), so if there is no convergent subsequence, then the sequence itsself cannot converge either.
So the absolute value won't converge.
I tried to do it with contradiction as follows:
Suppose that |s| converges. Then it has a convergent subsequence. Since |s| is also a subsequence of {s}, the convergent subsequence of |s| lies inside {s}. So {s} has a convergent subsequence. Contradiction.
Any clarification will be greatly appreciated.
CC
Here is the question:
Prove that if the sequence {s} has no convergent subsequence then {|s|} diverges to infinity.
To me, this seems so easy, but I'm having a really hard time putting it down in a rigorous manner.
My thoughts are:
every convergent sequence has a convergent subsequence (theorem in the book), so if there is no convergent subsequence, then the sequence itsself cannot converge either.
So the absolute value won't converge.
I tried to do it with contradiction as follows:
Suppose that |s| converges. Then it has a convergent subsequence. Since |s| is also a subsequence of {s}, the convergent subsequence of |s| lies inside {s}. So {s} has a convergent subsequence. Contradiction.
Any clarification will be greatly appreciated.
CC