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!