x is a continuous function on [0,1] with uniform topology.
The part I don't understand is second from last equation, where we have inequality of x(t) and x(0) with sum. How can I make sense of that equation, or prove it?
2026-04-01 01:40:40.1775007640
Relatively compact set theorem from Billingsley's Convergence of Probability Measure
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Since\begin{multline}x(t)=x(0)+\left(x\left(\frac tn\right)-x(0)\right)+\left(x\left(\frac{2t}n\right)-x\left(\frac tn\right)\right)+\cdots+\\+\left(x\left(\frac{nt}n\right)-x\left(\frac{(n-1)t}n\right)\right),\end{multline}all you have to do is to use the triangle inequality.