Let $a and $f: (a,d) \rightarrow \mathbb{R}$. Assume $f$ is uniformly continuous on $(a,c)$ and $(b,d)$. Prove that $f$ is uniformly continuous on the interval $(a,d)$.
My proof [EDITED]: Let $f$ be uniformly continuous on $(a,c)$ and $(b,d)$. So $\forall \epsilon_1 \exists \delta_1$ s.t. if $x_1,y_1 \in (a,c)$ and $|x_1 - y_1| < \delta_1$ then $|f(x_1)-f(y_1)| < \epsilon_1$ and $\forall \epsilon_2 \exists \delta_2$ s.t. if $x_2,y_2 \in (b,d)$ and $|x_2 - y_2| < \delta_2$ then $|f(x_1)-f(y_1)| < \epsilon_2$. From here, I'm not sure where to go.
Is my proof clear? Or is it missing some needed information? Thanks in advance for any suggestions or help.