# 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*}

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.
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.