The title pretty much says it all. I am a bit unsure about an exercise. I will just call the term A and B since it is easier for me to write it down.
I have that A converges in distribution to B.
Now I was given a term let's call it C and I have to show that term C converges in distribution to $hB$. I rewrote C to hA and thus used that since A converges in distribution to B and since h is a constant I have that hA converges in distribution to hB.
I am thinking that it is valid since it is one of the properties for limits that $ lim (s_n*k) = k*lim (s_n)$ where k is a constant.
Convergence in distribution means that $\Bbb{E}f(A) \to \Bbb{E}f(B)$ for all bounded continuous $f$.
Now, let $f$ be an arbitrary bounded continuous function. Then so is $g : x \mapsto f(hx)$. We thus have $$ \Bbb{E}f(hA)=\Bbb{E}g(A) \to \Bbb{E}g(B) = \Bbb{E}f(hB). $$ Since $f$ was an arbitrary bounded continuous function, we get $hA\to hB$ in probability.