- #1

Samuelb88

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## Homework Statement

Suppose [tex]S_n > 0[/tex] and [tex]r S_n > S_{n+1}[/tex], where

*r*is a constant such that

*0<r<1*. Show that [tex] \lim_{n\rightarrow +\infty} {S_n} = 0[/tex].

## The Attempt at a Solution

First, I'd like to know if my proof is correct.

**Proof**: Suppose [tex]S_n > 0[/tex] and [tex]r S_n > S_{n+1}[/tex], where

*r*is a constant such that

*0<r<1*. First note [tex]S_n[/tex] is decreasing since [tex]S_n > S_{n+1}[/tex] if [tex]r S_n > S_{n+1}[/tex]. Moreover, for every

*n*[tex]S_n > 0[/tex], so [tex]r S_n > S_{n+1} > 0[/tex]. Therefore the [tex] \lim_{n\rightarrow +\infty} {S_n} = 0[/tex].

Suppose my proof is correct. Then my question is what role does

*r*play in this problem? It seems to me, so long as [tex]S_n[/tex] is decreasing, and [tex]S_n > S_{n+1}[/tex], and for every

*n*[tex]S_n > 0[/tex], then the [tex] \lim_{n\rightarrow +\infty} {S_n} = 0[/tex]. Right?