Unbounded sequences.

Question

I was hoping to see if anyone could help me with a thought that I have stumbled upon.

Is there a sequence $a_n$ which is unbounded and its inverse is also unbounded?

I can't think of one myself and I am just looking for some help.

Thanks in advance

Answer 1

As Kavi is correctly suggesting in one comment, we can consider

$$a_n=\begin{cases} n &\text{ if } n\text{ is even }\\ \frac{1}{n} &\text{ if } n \text{ is odd.}\end{cases}$$

Clearly it is unbounded since the even terms are pushing it to infinity.

Its inverse, that we denote $b_n:=\frac{1}{a_n}$, is the same sequence but exchanging the parity of $n$ in the definition. More formally, we have that:

$$b_n=\begin{cases} n &\text{ if } n\text{ is odd }\\ \frac{1}{n} &\text{ if } n \text{ is even.}\end{cases}$$

Hence it is also unbounded.