# Inf A= 0 <=> 0 is an accumulation point

• Dec 6th 2009, 11:34 PM
mtlchris
Inf A= 0 <=> 0 is an accumulation point
Let A be a subset of the positive real. Prove that inf a = 0 <=> 0 is accmulation point of A

is it still true if A is the set of non negative reals? proof or give counter example

So I think I understand why this is true...since inf A=0 A is bounded below by 0, and we can have elements of A as close as we want to 0, but I dont know how to write it out?
• Dec 7th 2009, 02:51 AM
Plato
Quote:

Originally Posted by mtlchris
Let A be a subset of the positive real. Prove that inf a = 0 <=> 0 is accmulation point of A
is it still true if A is the set of non negative reals? proof or give counter example

Notice that if $\displaystyle c>0$ then $\displaystyle c$ is not a lower bound for $\displaystyle A$.
So $\displaystyle \left( {\exists x \in A} \right)\left[ {0 < x < c} \right]$.
In this way construct a decreasing sequence of distinct terms from $\displaystyle A$ converging to $\displaystyle 0$.

Consider this set of nonnegative numbers: $\displaystyle \left\{ 0 \right\} \cup \left\{ { 2 + \frac{1}{n}:n \in \mathbb{Z}^ + } \right\}$.