When does convergence in distribution imply convergence in probability? I was looking at the proof for the Delta Method (http://en.wikipedia.org/wiki/Delta_method#Proof_in_the_univariate_case) and there is something I am quite confused about.
It gives $\sqrt(n)[X_n - \theta]$ converges in distribution to $\text{N}(0,\sigma^2)$ where $\theta$ is some fixed value. It then says $X_n$ converges in probability to $\theta$. I can't figure out why this is true. 
Edit: (a solution using Slutzky's)
I've thought of another way to see why this is true using Slutzky's theorem.
$\sqrt(n)(X_n-\theta) \rightarrow N(0,\sigma^2)$ in distribution by assumption.
$1/\sqrt(n) \rightarrow 0$ and thus also converges to 0 in probability.
By Slutzky's, $(X_n-\theta) \rightarrow 0$*N(0,$\sigma^2$) = 0 in distribution.
Thus, $X_n-\theta \rightarrow 0$ in probability.
 A: 
When does convergence in distribution imply convergence in probability?

When the limit in distribution is a constant.
This is so in your case since one assumes that $\sqrt{n}(X_n-\theta)$ converges in distribution, hence $X_n-\theta$ converges in distribution to $0$, that is, $X_n$ converges in distribution to the constant $\theta$.
A: OK, $\sqrt(n) (X_n - \theta)$ converges in distribution to a normal distribution with expectation 0 and finite variance. Intuitively, since $\sqrt(n)$ goes to infinity, if $X_n - \theta$ did not converge in probability to zero,  the hole thing would blow up! Now, more formally, convergence in probability to zero means that for every positive $\epsilon$, there is an $N$ such that for all $n \ge N$ $P(|X_n - \theta| \ge \epsilon) \le \epsilon$. Now if this is not true, then for some positive $\epsilon$,  for all $n$ $P(|X_n-\theta| \ge \epsilon) > \epsilon$. 
Multiplying with $\sqrt(n)$ we get, that for all $n$,
$
     \sqrt{n}  |X_n- \theta| $ will be larger than $\sqrt{n} \epsilon$ with probability larger than $\epsilon$ for all $n$, with other words, some $\epsilon$-atom of probability disappears to infinity, contradicting convergence in distribution.
