- #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?