# Find the supremum and infimum of the following set

• Sep 12th 2011, 10:47 PM
My Little Pony
Find the supremum and infimum of the following set
Let A = {n^2/2^n | n in N}. Find the supremum and infimum, proving your assertions.

Attempt at Solution

SupA:

The highest term in the set is 9/8 and thus it is the supremum, by definition.

InfA:

The terms in the set approach 0 as n becomes arbitrarily large. It is clear that 0 < n^2/2^n for all n, thus, 0 is a lower bound.

I'm not sure how to prove that 0 is the infimum. Can someone lead me in the right direction?
• Sep 13th 2011, 02:20 AM
Plato
Re: Find the supremum and infimum of the following set
Quote:

Originally Posted by My Little Pony
Let A = {n^2/2^n | n in N}. Find the supremum and infimum, proving your assertions.
InfA:
The terms in the set approach 0 as n becomes arbitrarily large. It is clear that 0 < n^2/2^n for all n, thus, 0 is a lower bound.
I'm not sure how to prove that 0 is the infimum. Can someone lead me in the right direction?

You have already shown that 0 is a lower bound for the set.
Now show if $\displaystyle c>0$ then there is a $\displaystyle k\in\mathbb{N}$ such that $\displaystyle 0<\frac{k^2}{2^k}<c$.
Thus proving that no number greater that 0 is a lower bound.
That can be done by using the fact that $\displaystyle \left(\frac{n^2}{2^n}\right)\to 0.$