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

1. ## 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?

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

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