Question

Suppose {an}∞n=1 be a sequence of positive real numbers and 0 < x < 1. If an+1 < x·an for every n ∈ N, prove that limn→∞an = 0

Answer 1

Given that $a_{n+1} < x. a_n$ and $0<x<1$ we have that $a_{n+1} < a_{n} \forall n \in \mathbb{N}$ So as $(a_n)_{n=1}^{n= \infty}$ is a sequence of positive real numbers; we have that as n gets larger and larger $a_{n}$ goes to zero i.e. $\lim \limits_{n \to \infty} a_n = 0$