Proving/Disproving Sequence Claims: a Convergent Sequence Example

[kali0712]

This is a question on a recent assignment that I can't figure out. I think if I understood the first part, I could get the rest.

Let {a_n} be a convergent sequence with limit L. Prove or provide counter examples for each of the following situations. Suppose that there exists a number N such that:

a) a_n >/= 0 for all n>N; is it true that L >/= 0?

b) a_n > 0 for all n>N; is it true that L>0?

c) a_n > 0 for all n>N; is it true that L </= 0?

I know there should be a counterexample for a) but I just can't wrap my brain around it. Thanks a lot.

 

[AKG]

Do you have an actual question you're asking or do you just want someone to give you the answers?

Step 1) Use your intuition to guess whether the statements are true or false
Step 2) Try to prove your guesses
Step 3) If you can't prove one of your guesses, try proving the opposite
Step 4) If step 3 fails, go back to step 2.

 

[kreil]

If a sequence is bounded above or below, then that sequence has a limit and the limit is equal to that bound. Similarly, If a sequence has a limit then the sequence has a bound and the bound is equal to the limit.

I think this is relevant...

Josh

 

[AKG]

kreil, almost none of that is true. The sequence (1,0,1,0,1,0,1,0,...) is bounded above and below but has no limit. And saying "the limit is equal to that bound" doesn't even make sense because a sequence generally has infinitely many bounds, there is just a unique least upper bound or greatest lower bound, if they exist.