If I have an inequality, e.g.: $$\mathbb{E}|X(t+δ)-X(t)|²≤(aδ+bδ²)K$$ say $a,b$ and $K$ are just constants, $X$,an arbitrary stochastic processes, then if I want to evaluate the limit $lim_{δ→0}$: $$\lim_{δ→0}(aδ+bδ²)K=0$$ Is it correct to say then: $$\lim_{δ→0}\mathbb{E}|X(t+δ)-X(t)|²=0$$ i.e. convert the inequality to an equality like this. I was told it was incorrect to do this because the difference of two stochastic processes (say any arbitrary stochastic process) could be negative, but i don't see how this is possible since it is the magnitude squared. Then the only possible solution would be zero
(I've corected the Y, to X(t))and yes euclidean metric.