Assume $X_{n}$ be a sequence such that: $$ \sqrt{n}(X_{n} - X_{0})\stackrel{d}{\to} Z $$ Next, let $Y_{n}$ be a sequence of r.v. defined on the same probability space, such that $Y_{n} = X_{n} + o_{p}(1)$.
Is the following true? $$ \sqrt{n}(Y_{n} - X_{0})\stackrel{d}{\to} Z $$
No. Take $H_n = \frac{1}{\sqrt{n}}$ then certainly $H_n = o_p(1)$. However, when taking $Y_n = X_n + H_n$. We have that $\sqrt{n}(Y_n-X_0) = \sqrt{n}(X_n - X_0) + 1$ which converges in distribution to $Z + 1$.