# Bound and approximation of function of stochastic processes

I got two issues that seem very easy on first sight, but I got problems proving them. I have two pairs of stochastic processes $$\{X_{n,j}(t_j) : t_j \geq 0 \}$$ and $$\{Y_{n,j}(t_j): t_j \geq 0\}$$ for $$j=1,2,$$ and can suppose that for both $$j$$ they satisfy

$$\vert X_{n,j}(t_j) – Y_{n,j}(t_j) \vert \leq C_j t_j^{1/2 – \beta_j}$$ for some $$\beta_j > 0$$

and (under some more regularity conditions)

$$\sup \limits_{t_j \in [0,1]} \vert X_{n,j}(t_j) – Y_{n,j}(t_j) \vert = o(1)$$ as $$n \to \infty$$.

Now I want to verify if also $$\vert \sum_{j=1}^2 X_{n,j}^2(t_j) – \sum_{j=1}^2 Y_{n,j}^2(t_j) \vert \leq \sum_{j=1}^2 C_j t_j^{1/2 – \beta_j}$$

and

$$\sup \limits_{t_1,t_2 \in [0,1]} \vert \sum_{j=1}^2 X_{n,j}^2(t_j) – \sum_{j=1}^2 Y_{n,j}^2(t_j) \vert = o(1)$$ as $$n \to \infty$$

holds. This seems very simple at first, since I only use the continous function $$(x,y) \mapsto x^2+y^2$$ here, but the continous mapping theorem doesnt seem to be the correct way to prove this. Can anyone lead me into the right direction?

Thank you!