# question regarding contractive sequence

• Oct 9th 2009, 11:58 AM
dannyboycurtis
question regarding contractive sequence
Here is the problem:
Define the sequence {$\displaystyle a_n$} by $\displaystyle a_{n+1} = (a_n)^2$ for all $\displaystyle n\in\mathbb{N}$ where $\displaystyle 0<a_1\leq\frac{1}{3}$.
Prove that {$\displaystyle a_n$} is contractive.
Any advice and or solutions? thanks a lot
• Oct 9th 2009, 02:24 PM
tonio
Quote:

Originally Posted by dannyboycurtis
Here is the problem:
Define the sequence {$\displaystyle a_n$} by $\displaystyle a_{n+1} = (a_n)^2$ for all $\displaystyle n\in\mathbb{N}$ where $\displaystyle 0<a_1\leq\frac{1}{3}$.
Prove that {$\displaystyle a_n$} is contractive.
Any advice and or solutions? thanks a lot

What is a "contractive sequence", anyway?

Tonio
• Oct 9th 2009, 02:44 PM
Plato
Quote:

Originally Posted by tonio
What is a "contractive sequence", anyway?

I have wondered the same thing?
I know about Contraction mapping in a metric space.
Here is the only fact that I can relate it to: $\displaystyle 0 < a < 1\, \Rightarrow \,a^2 < a$.
• Oct 9th 2009, 04:36 PM
Krizalid