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## Homework Statement

Let ##(a_n)## be a arbitrary real sequence. Given that the sequence ##\frac{a_{n+1}}{a_n}## is convergent, show that ##\lim \frac{a_{n+1}}{a_n} = \lim \frac{a_n}{a_{n-1}}##

## Homework Equations

Take ##\mathbb{N} = \{1,2,3, \dots\}##

## The Attempt at a Solution

In general, I know that if ##(b_n)## is any convergent sequence then the limit of any subsequence is the same as the limit ##(b_n)##. So, for example, ##\lim_{k \to \infty} b_{k+1} = \lim_{n \to \infty} b_n##. However, I am not sure how to apply this to my current problem. It doesn't seem like ##\frac{a_k}{a_{k-1}}## is a subsequence of ##\frac{a_{n+1}}{n}## since the selection function would have to be ##n_k = k-1##, and this is not a map from ##\mathbb{N}## to ##\mathbb{N}##, since ##n_1 = 0 \not \in \mathbb{N}##.