Step function proof

Hello,

How would I show that:

If a < c < d < b, f is integrable on (a,b), then f is integrable on (c,d)?

We have already shown the following:

f is integrable on (a,b) $\Leftrightarrow \forall \varepsilon > 0{\rm{ }}\exists \varphi ,\psi {\rm{ (step functions) }}:{\rm{ }}\varphi \le f{\chi _{\left( {a,b} \right)}} \le \psi {\rm{ and }}I\left( {\varphi - \psi } \right) < \varepsilon$