Suppose I have processes $X_n(t)\overset{d}{\longrightarrow} X(t)$, and $Y_n(t)\overset{p}{\longrightarrow} ct$ for some constant $c$. Then, can I conclude like in Slutsky's theorem that $(X_n(t),Y_n(t))\overset{d}{\longrightarrow} (X(t),ct)$?
I tried to adapt the proof for normal random variables, so proving that for $\phi$ bounded and continuous, $\mathbb{E}[\phi(X_n(t),Y_n(t))]\to\mathbb{E}[\phi(X(t),ct)]$. First, we see that $\phi(X_n(t),ct)$ is also a bounded, continuous function, and by the convergence of $X_n(t)$, therefore $\mathbb{E}[\phi(X_n(t),ct)]\to\mathbb{E}[\phi(X(t),ct)]$, and thus $(X_n(t),ct)\overset{d}{\longrightarrow} (X(t),ct)$.
Also, $|(X_n(t),Y_n(t))-(X_n(t),ct)|\overset{p}{\longrightarrow}0$. Then, as here (Convergence in probability to a sequence converging in distribution implies convergence to the same distribution), we can show that $(X_n(t),Y_n(t))\overset{d}{\longrightarrow}(X(t),ct)$.
This all seems to go very well, but I am not completely sure whether my arguments actually hold for processes instead of random variables. So do you think this argument is correct? Or does an equivalent of Slutsky's theorem not hold for stochastic processes?