Follow from $d(X_n, Y_n) \rightarrow 0$ in probability that $\log(n)f_n(X_n) = 0 \implies \log(n)f_n(Y_n) = 0$ Lets consider two sequences of random variables $X_n, Y_n$ with values in a metric space $(M, d)$ and $d(X_n, Y_n) \stackrel{\mathbb{P}}{\longrightarrow} 0$. Now suppose there is a sequence $f_n$ of continuous functions, such that $\ln(n)f_n(X_n) \stackrel{\mathbb{P}}{\longrightarrow} 0$. Can we conclude at this point, 
$$\ln(n)f_n(Y_n) \stackrel{\mathbb{P}}{\rightarrow} 0$$
I think it is true, that $f_n(Y_n)$ will converge to zero in probability, but will it still shrink fast enough to catch up with the $\ln(n)$ term?

It is false.
Consider $f_n(x) = nx$, and suppose $X_n$ and $Y_n$ are constant functions $X_n \equiv 0$, $Y_n \equiv 1/n$. You have that $d(X_n,Y_n) \to 0$, and $\ln(n)f_n(X_n) = 0$, but $\ln(n)f_n(Y_n) = \ln(n)$ that does not converge to zero.
