Context and assumptions : In an exercice , i consider a couple of reals (a,b) such that $a\neq b$, and $a\leq b$. we choose these two reals to be such as $\left]a,b \right[\subset \left[0,1\right]$
In a first time , we say that there exists an integer $N$ such that for each $n\geq N$, $\vert ln(n+1)-ln(n)\vert < b-a$
Let's then fix this integer $N$ (I mean , it exists , and let's call it $N$)
we then consider $k$ , an integer such as $ln(N)-k \leq a$
Question : Show that there exists an integer $n \geq N+1$ such that $ln(n)-k \in \left] a,b \right[$.
what i have tried : I thought using the absolute value to create inequality was the obvious way of unwinding the problem , more precisely i tried to manipulate this inequality : $$a-b\leq ln(N+1)-ln(N) \leq b-a $$ I tried adding $k$ , using it's properties , but i can't solve it properly..
Also , i'm conscious the title for the question is a bit "wide" in sense , so feel free to change it if you have a better idea.