Uniform Continuity of functions and their restrictions

Question

Suppose $f$ is a function mapping a set $S$ to $\mathbb{R}$. $A \subseteq S$. If $f$ is uniformly continuous on $S$, how would I show that the restriction of $f$ to $A$ is uniformly continuous?

Answer 1

Just verify it by definition. For a given $\varepsilon>0$, the same $\delta$ will work as for $f$.

Answer 2

Write the definition of the uniform continuity of $f$ and just in $$\forall \epsilon>0.....,\forall x,y\in S,|x-y|<\delta..............$$ replace $S$ by $A$.