If a function $f$ is integrable on $[a,b]$, prove that it is integral on $[a,c]$ and $[c,b]$ and that $\int_a^b f = \int_a^c f + \int_c^b f$

Question

I am having a lot of trouble with this question. I think that I am meant to set up 3 partitions, $P,\; \alpha\;\text{and} \; \psi$, where $P={\{x_0, x_1, \cdots,x_n\}}$, $\alpha={\{x_0, x_1, \cdots,x_k\}}$ and $\psi={\{x_k, x_{k+1}\cdots,x_n\}}$. Then I attempted to use the integrability criterion that for an arbitrary $\epsilon > 0$ we have

$$ U(f,P) - L(f,P) < \epsilon $$ and then I think I am meant to use that inequality with the fact that $U(f,\alpha)\leq U(f,P)$ and that $U(f,\psi)\leq U(f,P)$. But this is where I lose track. I have read some things also about the infinimum and supremum but I do not understand their applications in this question.