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## Homework Statement

Suppose {xn} is a sequence of positive numbers for which there exists c, 0<c<1, such that ([x][/n+1])/([x][/n])<c for all n in Z+. Prove that [x][/n] goes to zero.

## Homework Equations

## The Attempt at a Solution

Let the first term of {xn} be x, where n=1. Then by the given, [x][/n+1]/[x][/n]<1, therefore, [x][/1]>[x][/2]>[x][/3]>...>[x][/n]>[x][/n+1], hence sup{[x][/n]} = [x][/1].

By the given, inf{[x][/n]}=0 so {xn} is bounded and strictly decreasing. We know a monotone sequence converges if and only if it is bounded, but I am having trouble proving that {xn} goes to zero.