Question 2
Let $f:\mathbb R\to\mathbb R$ be a nondecreasing function such that $f(y)-f(x)\lt y-x$ for all real numbers $x,y$ with $y\gt x$. Let $(u_n)_{n\geq0}$ be a sequence satisfying
$$
u_{n+2}=f(u_{n+1})-f(u_n)
$$
for every integer $n\geq0$.
Prove that $(u_n)_{n\geq0}$ converges to $0$, that is, for every real $\varepsilon\gt0$, there is an integer $N$ such that $|u_n|\leq\varepsilon$ for every integer $n\geq N$.
$$
u_{n+2}=f(u_{n+1})-f(u_n)
$$
for every integer $n\geq0$.
Prove that $(u_n)_{n\geq0}$ converges to $0$, that is, for every real $\varepsilon\gt0$, there is an integer $N$ such that $|u_n|\leq\varepsilon$ for every integer $n\geq N$.