I am currently self studying Riemann integrability and got to a part that states that if partitions P and Q are such that Q is a refinement of P then $$L (f,P) \leq U (f,Q) \leq U (f,Q) \leq U (f,P)$$ I fully understand the proof. It is based on the fact that every point of P is in Q.
My thought is this: Suppose I have a partition dependent on n, say: $$P_n:=\{x_0,x_{1,n},x_{2,n},...,x_{p,n}\}$$ Like for example, $$P_n:=\{0,\frac{1}{n},\frac{2}{n},...,1\}$$ Will I also have that $$L (f,P_n) \leq U (f,P_{n+1}) \leq U (f,P_{n+1}) \leq U (f,P_n)?$$ I am stuck on how to approach this, or even get a counter example.
If it is false, should there be any condition to be put on $f$, say continuity or something else?