I have the following problem.
Let $X_n \sim \operatorname{Beta}(1/n,1/n)$ be a sequence of beta distributions, and $X \sim \operatorname{Bin}(1,1/2)$ which is equivalent to $\operatorname{bern}(1/2)$. Show that $X_n \leadsto X$, i.e., $X_n$ converges in law (or distribution) to $X$.
What I have to prove is that $F_{X_n}(x) \rightarrow F_{X}(x)$ for all continuity points $x$ of $F_X$. Now $$ F_{X}(x)= \begin{cases} 0 & x < 0 \\ \frac{1}{2} & 0 \leq x < 1 \\ 1 & x \geq 1 \end{cases}. $$ I already know the cdf of a beta distribution
$$ F_{X_n}(x)= \begin{cases} 0 & x < 0 \\ I_{x}\left(\frac{1}{n},\frac{1}{n}\right) & 0 \leq x < 1 \\ 1 & x \geq 1 \end{cases}, $$ where $I_{x}\left(\frac{1}{n},\frac{1}{n}\right)$ is the regularized beta function $\frac{B(x,1/n,1/n)}{B(1/n,1/n)}$. But now I can't figure out how to complete the convergence on the interval $(0,1)$ because the expresion for $I_{x}\left(\frac{1}{n},\frac{1}{n}\right)$ has a lot of terms that doesn't seem to go to $1/2$. Any hint? Thanks
Since $X_n$ has characteristic function ${}_1F_1(\tfrac1n,\,\tfrac2n;\,it)$, we wish to show this function has $n\to\infty$ limit $\frac{1+e^{it}}{2}$. Going by powers of $it$, we wish to show $\lim_{n\to\infty}\frac{(\tfrac1n)_k}{(\tfrac2n)_k}=\frac{1+\delta_{k0}}{2}$, where $(x)_k$ is a falling Pochhammer symbol. The cases $k=0,\,k\ne0$ are easily separately checked.