why $t_n\to 0$.

Question

Let $(t_n)$ and $(v_n)$ be two sequence such that $\forall n\in\Bbb N: 0\leq t_n\leq 1$ and $v_n\geq 0$ with $v_n\to 0$. Assume $\forall n\in\Bbb N: 0\leq t_{n+1}\leq \frac{t_n+v_n}{M}$ where $M>1$. Why $t_n\to 0$.

Thank you in advance

Answer 1

$0 \leq t_{n+1} \leq \frac{t_{n}+v_{n}}{M}$. Taking the limit as $n\to \infty$, we get

$0 \leq \lim_{n\to \infty}{t_{n+1}} \leq \lim_{n\to \infty}\frac{t_{n}+v_{n}}{M} \implies 0 \leq \lim_{n\to \infty}{t_{n+1}} \leq \lim_{n\to \infty}\frac{t_{n}}{M} $

Now, notice that, $\lim_{n\to\infty} t_{n+1}$ must equal $\lim_{n\to\infty} t_{n}$. Call it L. So,

$0\leq L \leq \frac{L}{M}$. Which for $M>1$ is possible if and only if $L=0$ as desired.