Prove or Disprove: For all $x \in \mathbb{R}$, there exists a positive real number$\epsilon$ such that either $x \leq (-\epsilon) or x \geq \epsilon$ Can you guys help me prove or disprove this?
Also this is what I got for the negation of this statement: There exists $x \in \mathbb{R}$ for all positive real numbers $\epsilon$ such that either $x &gt; (-\epsilon)$ or $x &lt; \epsilon$

Is my negation correct?
Where should I start with my proof?
Your negation is not quite correct, because the negation of 


  either $x\le-\epsilon$ or $x\ge\epsilon$


is not


  either $x&gt;-\epsilon$ or $x&lt;\epsilon$;


it’s


  $x&gt;-\epsilon$ and $x&lt;\epsilon$.


However, the original statement is not true: it has one (and only one) exception. What real number is that exception?
Let $x \in R$. 

Let $\epsilon \in R : 0 &lt; \epsilon &lt; x$ or $ \epsilon &lt; -x &lt; 0$.

Since $x + \epsilon \in R$, and $0 &lt; \epsilon$, your statement is true.