This is part of some exercises we got to prepare ourselves for an exam in two weeks. It isn't homework. What are counter-examples to parts 1 and 3? How can we prove part 2?
Let $a_n,b_n$ be sequences such that $ |a_n - b_n | \to 1 $ .
1) If $a_n \to L <\infty $ then $ b_n $ also converges to a finite limit.
2) If $ a_n $ is bounded then $b_n $ is also bounded.
3) If $ a_n $ is monotone increasing, then $b_n $ is also monotone increasing.