Are the following definitions of a uniform continuous variable equivalent? We say $f$ is uniformly continuous on a set $D$ if 
\begin{align*}
\forall \epsilon >0, \exists \delta >0 , s.t.\\
\forall x, a \in D, |x-a| \le \delta \rightarrow |f(x)-f(a)| \le \epsilon  
\end{align*}
Is this definition equivalent to the following:
\begin{align*}
\forall \epsilon >0, \exists \delta >0 , s.t.\\
|x-a| \le \delta \rightarrow \sup_{x, a \in D}|f(x)-f(a)| \le \epsilon  
\end{align*}
 A: The second definition uses the notation 'sup' meaning supremum or least upper bound.  It is quite possible to define uniform continuity using this notation, for example:
\begin{align*} \forall \epsilon >0, \exists \delta >0 , s.t.\sup_{x, a \in D s.t. |x-a| \le \delta }|f(x)-f(a)| \le \epsilon \end{align*}
This is equivalent to the first definition. However, in the second definition the expression $|x-a| \le \delta$ is neither within quantifiers over $x$ or $a$ nor within the scope of 'sup'.  Taken strictly, there is nothing in the second definition to say that the $x$ and $a$ in that expression belong to $D$.  So although the second definition is not incorrect or inconsistent with the first definition, its use of notation is not as clear as it could be.
A: They are the same, because if the 'sup' (an upper bound, specifically the least upper bound) of $|f(x) - f(a)| < \epsilon$ as in your definition, then all $|f(x) - f(a)|$ must be less than $\epsilon$, which is equivalent to the definition of uniformly continuous.
