Quadratic variation from completed grid increments ($S_n(t)={\sum }_{t_k^n\le t}(X_{t_k^n}-X_{t_{k-1}^n}{)}^2$)

Completed squared increments form a nondecreasing step process. The continuous sum that includes the last partial increment differs by at most the square of the path's modulus of continuity over one mesh interval. Therefore both have the same limit in uniform convergence on compacts in probability. This proves monotonicity of the limiting quadratic variation without incorrectly asserting monotonicity of each continuous partial-increment sum.