- #1

DeadOriginal

- 274

- 1

I have a quick question about limits. This is something I should know but shame on me I forgot.

If a function is bounded both above and below but isn't monotonic and is not necessarily continuous at all points, how do I go about proving its limit exists?

In particular I am thinking about something along the lines of this function:

$$

\lim\limits_{n\rightarrow\infty}\frac{F_{n+1}}{F_{n}}

$$

where $$F_{n}$$ is the nth Fibonacci number. I know that the limit exists and I know what that limit is but say I didn't. Say I wanted to prove that it did exist. How do I do that without using the closed form expression for Fibonacci numbers? I know that it is bounded but it isn't monotonic.

This wasn't a homework question but a question that came to my mind after proving that this limit was equal to the golden ratio using the closed form expression of the Fibonacci numbers.