Analysis Applying Combination Rules

[nlews]

True or False, with a proof or counterexample.

a) If bn ≠ 0 and an/bn →1, then an-bn → 0
b) If bn ≠ 0, bn is bounded and an/bn → 1 then an-bn → 0

At the moment I cannot even see which is false so I am struggling with this question. I think the proof will require use of the quotient combination rule and the sum combination rule but I cannot see where to start! Any help would be appreciated!

 

[rochfor1]

Do you at least have a guess?

 

[nlews]

Yes! Sorry I am new so wasnt sure how this all worked!

Basically I think the first is false and the second true!
I think this because for the second one,
if i use the fact that an/bn →1
I can say that
|an-bn| = bn |an/bn -1| → 0 because bn is bounded.

I think this works?

I can't really prove that the first is false, but I am working on that one at the moment! Any help would be great!

 

[rochfor1]

Yes. That absolutely works for the second. For the first consider a_n = n + 1 and b_n = n.