The above trick was used to solve the following question without explanation. Could someone prove it? Question: $X_1, X_2,...$ and $Y_1, Y_1,...$ are two sequences of random variables such that $$ \sqrt n (X_n -a)\xrightarrow{\text{d}} N(0,b) $$ $$ \sqrt n (Y_n -c)\xrightarrow{\text{d}} N(0,d) $$ for some a,b,c and d, and c $\ne 0$ Prove that sequence of random varilables $Z_n$ defined as $$Z_n=\sqrt n \frac{X_n-a}{Y_n}$$ converge in distribution to a normal distribution. Find the parameters of the normal distribution.
Solution: c = plim($Y_n$) $$Z_n=\sqrt n \frac{X_n-a}{c+o_p(1)}$$
Since $\frac{1}{c+o_p(1)} = \frac{1}{c} + o_P(1)$ and $ \sqrt n (X_n -a)\xrightarrow{\text{d}} N(0,1) $. Then
$$Z_n = \sqrt n (X_n -a) (\frac{1}{c} + o_p(1)) = \frac{\sqrt n (X_n -a)}{c} + o_p(1)\sqrt n (X_n -a) =$$ $$\frac{\sqrt n (X_n -a)}{c} + o_p(1)O_p(1) \xrightarrow{\text{d}} N(0,\frac{b}{c^2}) $$
I solved the question using Slutsky and did not have to use Oh Pe notation. I understand all parts of the solution apart from the above-mentioned trick.