Let { } be a sequence and k be a positive integer.
a) If { } converges to L, what are the limits of { } and { }? Why?
b) If { } diverges, what is the limit of { }? Why?
There's a theorem somewhere to the effect that if a sequence tends to a limit, then any subsequence of that sequence tends to the same limit. Let me go away and find it ...
Aha, found it:
http://www.proofwiki.org/wiki/Limit_of_a_Subsequence
... work in progress to expand it to make it more general, but it's there for the real number plane.
May be the following fill be helpful.
Just think that you walk from the point to the point .
And suppose that there is no other path is possible, i.e., you go on a unique path, and while walking mark your steps with numbers, step , step , and so on.
Say you have reached to after steps.
Now, can you tell me, could you reach at without making steps if you had chance to start from the point that you made step ?
Or if you had enough long legs, could you jump from the place of step to the place of step , then to the place of step , and so on, to reach ?
What you wrote above is exactly the same, as , and what happens if I ignore the first terms? I will still be able to reach to by running on the remaining terms of the sequence...
I think it is now more clear, let me know if not...