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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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    @yuta, well, do you have a def$i$n$i$tion for convergence? What have you trie$d$ 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: $(n+(-1)^n)_{n\in \mathbb{N}}, \ \ \ (n)_{n\in \mathbb{N}} ,\ \ (x_n)_{n\in \mathbb{N}} \ \text{with} \ x_n=0 \ \forall n, \ \text{and} \ \ ((-1)^n)_{n\in \mathbb{N}}.$