Question about convergence of a sum of integrals as mesh tends to $0$ using continuity

Question

Below is a proof on approximating stochastic integrals for mean-square continuous processes in René Schilling's Brownian Motion.
My question is: How do we get the last convergence to $0$ in the proof? I.e. why is
$$sum_{j=1}^n int_{s_{j-1}}^{s_j} sup_{u,v in [s_{j-1},s_j]} E[|f(u)-f(v)|^2] dt to 0$$ given that we have $lim_{s to t} E[|f(s,cdot)-f(t,cdot)|^2 ] = 0$ for each $t in T$?
I am confused because the assumption only gives continuity at a point, i.e. $lim_{n to infty} E|f(s_n)-f(s)|^2=0$ as $s_n to s$. How do we get $sup_{u,vin[s_{j-1},s_j]} E[|f(u)-f(v)|^2] to 0$ as $|Pi| to 0$?
Also from this we have $sum_{j=1}^n int_{s_{j-1}}^{s_j} sup_{[s_{j-1},s_j]}E|f(u)-f(v)|^2 dt$ of the form $sum_j (s_j - s_{j-1}) g_j$ where $g_j to 0$ as $|Pi| to 0$. So we could bound this by $|Pi| sum_{j=1}^n g_j$, but we how do we get the sum tend to $0$ when each $g_j$ does? I would greatly appreciate any help resolving these issues.

Stephen Montgomery-Smith · Answer

Mimic any proof that a continuous function on a closed interval is uniformly continuous, to show that given $epsilon>0$, there exists $delta > 0$ such that $|s-t|<delta Rightarrow mathbb E[|f(s)-f(t)|^2] < epsilon$.

(An example of such a proof: suppose for some $epsilon>0$ there is no such $delta$>0. Find a sequences $(s_n, t_n) in [0,T]^2$ with $|s_n - t_n| to 0$ and $mathbb E[|f(s_n)-f(t_n)|^2] > epsilon$. Let $(s,t)$ be a cluster point of ${(s_n,t_n)}$. So $s = t$. Since $mathbb E[|f(s_n)-f(s)|^2] to 0$ and $mathbb E[|f(t_n)-f(t)|^2] to 0$, it must be that $mathbb E[|f(s_n)-f(t_n)|^2] to 0$, which is a contradiction.)

So, if $|Pi| < delta$, then $$ sum_{j=1}^n int_{s_{j-1}}^{s_j} sup_{u,vin[s_{j-1},s_j]} mathbb E[|f(u)-f(v)|^2] , dt le sum_{j=1}^n int_{s_{j-1}}^{s_j} epsilon , dt = epsilon.$$

Question about convergence of a sum of integrals as mesh tends to $0$ using continuity

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