Weak convergence of discrete random variables to a continuous variable The question is from Billingsley. $X_n \in \{\gamma_n+k\delta_n; k\in N\}, \delta_n>0$. Suppose $\delta_n\rightarrow 0$ and $k_n$ is an integer varying with n s.t. $\gamma_n+k_n\delta_n\rightarrow x$ and $P(\gamma_n+k_n\delta_n)\delta_n^{-1}\rightarrow f(x)$ where $f$ is the density of a random variable $X$. Show that $X_n\rightarrow X$ weakly.
My attempt: $X_n\rightarrow X$ weakly $\iff \int gd\mu_n\rightarrow \int gd\mu $ where $g$ is continuous, bounded and I think it is enough to be compactly supported. $\mu_n$ is the distribution of $X_n$.  
So need to show $\sum_n \delta_ng(\gamma_n+k\delta_n)P\{X_n=\gamma_n+k\delta_n\}\delta_n^{-1} \rightarrow \int g(x)f(x)dx$ which I guess follows from Riemmanian sum. Since g is bounded, we can move the limit inside but I am having trouble transitioning from $k$ to $k_n$ because we know $g(\gamma_n+k_n\delta_n)\rightarrow g(x)$ and $P\{X_n=\gamma_n+k_n\delta_n\}\delta_n^{-1} \rightarrow f(x)$ i.e. we have convergence for a particular sequence $\{k_n\}$. We need to show convergence $\forall k$, don't we? Any ideas? and thanks!

 A: You've phrased one of the conditions slightly incorrectly. As stated, it is unclear what the variable $x$ has to do with anything. The condition should read something like: "Suppose that whenever $k_n$ is a sequence of integers with $\gamma_n+k_n\delta_n\to x$, then $P[X_n=\gamma_n+k_n\delta_n]\delta_n^{-1}\to f(x)$".
With this in mind, here is a sketch to prove the assertion. First replace the discrete $X_n$ with a "histogrammed" version, call it $X'_n$, which for each integer $k$ places mass $P[X_n=\gamma_n+k\delta_n]$ uniformly over the interval of width $\delta_n$ centered at $\gamma_n+k\delta_n$. (Since the density of $X'_n$ is constant over the interval, this constrains the density's height over that interval to be...) You then prove that $X'_n$ converges in distribution to $X$ (applying the result of the exercise directly previous to this one) by showing that the density of $X'_n$ converges pointwise to the density of $X$. To show pointwise convergence, let $x$ be fixed. There exists for each $n$ an integer $k_n$ such that $x$ lies in the interval centered at $\gamma_n + k_n\delta_n$. Argue that this defines a sequence $(k_n)$ for which $\gamma_n + k_n\delta_n\to x$, and the result follows.
Having demonstrated this, you next write the original $X_n$ as $h_n(X'_n)$ for a suitable sequence of functions $h_n$ that converge pointwise to the identity function $h(x):=x$. Then argue that $h_n(X'_n)$ converges in distribution to $h(X)$. (See the exercise two previous to this one.)
