To prove the Upper Riemann Integral $\geq$ Lower Riemann Integral

Your first steps show that $L(P,f) \leqslant U(P,f)$, where the same partition $P$ is used in the lower and upper sum. To finish you need to show that for (different) partitions $P$ and $Q$ we also have

$$\tag{*}L(P,f) \leqslant U(Q,f)$$

It then would follow that with $Q$ fixed,

$$\sup_P L(P,f) \leqslant U(Q,f),$$

and, subsequently,

$$\sup_P L(P,f) \leqslant \inf_QU(Q,f)$$

To prove (*) take a common refinement $R = P \cup Q$ and show that we must have

$$L(P,f) \leqslant L(R,f) \leqslant U(R,f) \leqslant U(Q,f)$$

I'll leave that to you with the hint that you should consider what happens to the ordering of lower and upper sums when a single new point is added to a partition.