Given that $K > 1$. And that $\lim \epsilon_n = 0 $ such that $ \epsilon_n > 0 $. Also $ u_n \in [0,1]$, such that $ 0 \leq u_{n+1} \leq \frac{u_n + \epsilon_n}{K}$.
Prove that $\lim u_n = 0$.
There is an incredibly complicated solution using epsilons and summations in my textbook, but I thought of something much simpler.
Just take the limit of $ \lim 0 \leq \lim u_{n+1} \leq \lim \frac{u_n + \epsilon_n}{K}$, we get $ 0 \leq l \leq \frac{l}{K}$, such that $\lim u_n = l$. But $l \leq \frac{l}{K}$ implies $l \times( K - 1 )\leq 0$, since $u_n \geq 0$ and $ K > 1$, this clearly implies that $l=0$.
Is this solution correct?