In the problem I am working on the author wants me to show an example, or a case given various theorems that an example is impossible for the following problem:
An unbounded sequence $(a_n)$ and a convergent sequence $(b_n)$ with $(a_n – b_n)$ unbounded.
After some work trying to find an example, I figured this is likely not possible. I set off to prove it. Writing down what I know:
- A convergent sequence is bounded
I tried to prove it and ended up getting no where. I had attempted to show that $|a_n| \le M$ and we know $|b_n – b| \lt \epsilon$, but this ended up getting me no where either. The author's proof was elegant, but I am not sure how he can make such a conclusion so quickly:
Such a request is impossible. By Theorem 2.3.2 $(b_n)$ is bounded. If $(a_n – b_n)$ were bounded we could show that $(a_n) = (a_n – b_n) + (b_n)$ would also have to be bounded, which is not the case. So this is unbounded.
I am not sure how he jumped to the conclusion he did. Is there a rule I am missing? How can he conclude that in general $(a_n – b_n) + (b_n)$ could not happen given $(a_n)$ being unbounded and $(b_n)$ being convergent? In this section we didn't discuss the results of combining these types of sequences. Is there some place I can read more about these properties?
Best Answer
Suppose that $(c_n)_{n\in\mathbb N}$ and $(d_n)_{n\in\mathbb N}$ are bounded sequences. Then there are numbers $M$ and $N$ such that$$(\forall n\in\mathbb{N}):\lvert c_n\rvert<M\text{ and }\lvert d_n\rvert<N.$$But then$$(\forall n\in\mathbb{N}):\lvert c_n+d_n\rvert\leqslant\lvert c_n\rvert+\lvert d_n\rvert<M+N.$$That is, the sequence $(c_n+d_n)_{n\in\mathbb N}$ is bounded.