I have the following simple question but I am a bit confused and would appreciate some help.
Note: I denote $ P_n $ as the partition of the interval $ [a,b] $ and $ m_i $ as the least upper bound of $ f(x) $ over the sub-intervals $ [x_{i-1}, x_i] $.
If the function $ f : [a, b] → \mathbb{R} $ is bounded, Riemann integrable and satisfies $ f(x) \geqslant 0 $ for all $ x \in [a, b] $, show that $ \int_a^b f(x) \,dx \geqslant 0 $.
I understand that since $ f(x) \geqslant 0 $ for all $ x \in [a, b] $, means that the value of the greatest lower bounds, $ m_i $, are all greater than or equal to zero. And so the lower integral of $ f(x) $ (i.e., $ L(P_n,f) = \sum_{i=1}^n m_i(x_i-x_{i-1}) $) will be greater than or equal to 0 if $ \sum_{i=1}^n (x_i-x_{i-1}) \geqslant 0 $.
If I can say that $ L(P_n,f) \geqslant 0 $ then it's easy to show $ \int_a^b f(x) \,dx \geqslant 0 $, this is what I am interested in showing.
But given only the information in the question, how can we ensure that it's true that $ \sum_{i=1}^n (x_i-x_{i-1}) \geqslant 0 $?
For example, if we are told that $ f(x) $ is increasing, then I have no problem.
Any suggestions appreciated, thanks!