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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.

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    Since you seem to know that the first and third cases are not necessarily true, and that the second is, you must have some thoughts on it as to when and why/why not?2012-11-23
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    No... we actually have only the "true/false" answer... And I cant figure out how to build a counterexample2012-11-23
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    @yuta, well, do you have a definition for convergence? What have you tried to prove the statements?2012-11-23

2 Answers 2

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If $(a_n)$ is bounded, there is an $M>0$ with $|a_n| for all n. Now since $|a_n-b_n|\to 1$, there is a positive integer $N$ with $|a_n-b_n|<2$ for all $n>N$. Hence for all $n>N$, we have $|b_n|\leq |b_n-a_n|+|a_n|. Hence $(b_n)$ is bounded by $\max(M+2,|b_1|,...,|b_N|)$.

Let $a_n=0$ and $b_n=(-1)^n$ for counterexamples to the other parts.

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Hint:

For (2) use: $|a_n-b_n| \to 1 \Rightarrow b_n-a_n<|a_n-b_n|<2 \Rightarrow b_n for sufficiently large $n$.

For (1) and (3) use the sequences: $$(n+(-1)^n)_{n\in \mathbb{N}}, \ \ \ (n)_{n\in \mathbb{N}} ,\ \ (x_n)_{n\in \mathbb{N}} \ \text{with} \ x_n=0 \ \forall n, \ \text{and} \ \ ((-1)^n)_{n\in \mathbb{N}}.$$