EDIT : I have doubts about the $1_+$ notation so I'll come back here as soon as I got the answer. In the mean time please consider this question on hold. Feel free to comment if you have any inputs on the $1_+$ notation, thanks.
First of all I want to let you know that this is an homework assignment I was given.
So here is the question :
Given $(u_n)_n$ a sequence of strictly positive reals so that $\lim_{n\to\infty} {u_{n+1}\over u_n} = 1_+ $
Prove that $\lim_{n\to\infty} u_n$ exists and is $\ne 0$
Now I tried to use the limit and build from there but to no avail :
$$ \forall \epsilon >0 , \exists N \in \mathbb{N} : n>N \to 1\le{u_{n+1}\over u_n} \le 1+\epsilon $$
Please provide with a few hints to get me started, thanks a lot !
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