# Find sequences such that …

Let $$c\in \mathbb{R}$$. Find two sequences $$(a_n)_n$$, $$(b_n)_n \subset \mathbb{R}$$ with:

(i) $$\lim\limits_{n\to\infty}a_n=\infty, \lim\limits_{n\to\infty}b_n=0$$ and $$\lim\limits_{n\to\infty}a_nb_n=c$$

My example:

$$a_n:=n$$ and $$b_n:=\frac{1}{n}$$

(ii) $$a_n \neq 0 \neq b_n$$ for all $$n\in \mathbb{N}$$, $$\lim\limits_{n\to\infty}a_n=0=\lim\limits_{n\to\infty}b_n$$ and $$\lim\limits_{n\to\infty}\frac{a_n}{b_n}=c$$

My example:

$$a_n:=\frac{1}{n^2}$$ and $$b_n:=\frac{1}{n}$$

(iii) $$\lim\limits_{n\to\infty}a_n=\infty, \lim\limits_{n\to\infty}b_n=-\infty$$ and $$\lim\limits_{n\to\infty}(a_n+b_n)=c$$

My example:

$$a_n=n$$ und $$b_n=-n$$

Are these valid examples?

• Hmm, don't I want that? – Analysis May 26 '19 at 20:06
• My error, I thought the examples all referred to the first question. Your examples are all correct. – lulu May 26 '19 at 20:07
• It seems your examples are forgetting about the constant $c$. For instance, your example for (i) only considers $c=1$. – Dave May 26 '19 at 20:12
• ... which is an element of the real numbers, what's your point? – Analysis May 26 '19 at 20:13
• My interpretation of the questions is that you want the limit to equal the parameter $c$. – Dave May 26 '19 at 20:14

These are so very close to being correct. The thing is, for example in part (i), the limit is

$$\lim_{n\to\infty}a_nb_n=\lim_{n\to\infty}1=1\neq c$$

How do you fix this? Well, you make $$a_n=n$$ as usual, but $$b_n\frac{c}{n}$$ so that their product is always $$c$$.

For part (ii), your answer isn't even correct -- the limit $$lim_{n\to\infty}\frac{a_n}{b_n}=0\neq c$$.

Basically the intuition here is we sort of want $$a_n$$ to be "c times" $$b_n$$ -- so if $$a_n=\frac{c}{n}$$ and $$b_n=\frac{1}{n}$$ then we're fine.

This is very similar to before -- your limit is $$0$$. You want $$a_n=n+c,b_n=-n$$ to ensure the limit is $$c$$ not $$0$$.

Pedantic note: When I write "$$\neq c$$" what I'm really saying is, there exists $$c$$ such that the two are unequal.

• :-) Understood! – Analysis May 26 '19 at 21:36

Your examples are correct; you could generalize your $$b_n$$ to be $$b_n=\frac{c}{n}$$ for any fixed $$c\in\mathbb R$$. In this case, you'll satisfy all of the conditions you want.