EDB — 19S

[19S] Prerequisites:[19Q]↺↻, [1HS]↺↻.

Let \(I⊂ ℝ\) be an interval with extremes \(a,b\). Let \(f,f_ n:I→ℝ\) be continuous non-negative functions such that \(f_ n(x)↗_ n f\) pointwise (i.e. for every \(x\) and \(n\) we have \(0≤ f_ n(x) ≤ f_{n+1}(x)\) and \(\lim _ n f_ n(x) =f(x)\)). Prove that

(Note if the interval is open or semiopen or unbounded then the Riemann integrals are understood in a generalized sense; in this case the right term can also be \(+∞\)).

The previous result is called Monotonic Convergence Theorem and holds in very general hypotheses; in the case of Riemann integrals, however, it can be seen as a consequence of the results [19Q]↺↻ and [1HS]↺↻.