I am unable to understand the following applications of Dini's theorem. In particular, verification of continuity of the sequence of functions.
To prove the Riemann sum uniformly converges to the integral of a continuous function? Specifically this question When does a Riemann sum converge uniformly?. While I can understand the accepted answer, I am not sure how we can use Dini's theorem as mentioned in the second response since we have to check the "continuity" of the Riemann sums for it to be valid.
Similarly, In John Walsh's Intro to Probability book. The author proves the following theorem (a similar discussion here which I did not understand clearly: A strange proof of Glivenko-Cantelli with Dini) .
(Glivenko-Cantelli Theorem): Let $X_1, X_2, \ldots $ be a seq. of i.i.d. r.v.'s with common distribution $F$. Define the empirical distribution function $F_n(x)$ by $$F_n(x) = \frac{1}{n} \sum_{i=1}^n \mathbf{1}_{\{X_j \leq x\}}$$
Then $F_n$ converges uniformly to $F$ a.s. as $n \rightarrow \infty$
Proof: Let $U_1, U_2, \ldots $ be a seq. of i.i.d Unif(0,1) r.v.'s and let $F^0$ be the the Unif(0,1) distribution, and let $F_{n}^0$ be the empirical distribution function. The author states that $F^0$ and $F_n^0$ are monotone and $F^0$ is continuous, so by Dini's theorem, the seq. $\{F_n^0\}$ converges uniformly to $F^0$ for every x.
While the monotonicity and the pointwise convergence conditions of Dini's Theorem are clear to me in these two examples, I am unable to understand why the continuity of the seq. of functions is obvious. Is there any common principle that I am missing here?