\section{Introduction}
Let $X_1,\cdots,X_n$ be a random sample of size n from a random
variable $X$ where $X$ has finite variance $\sigma^2$ and let
$S^2=\frac{1}{n-1}\sum_{i=1}^{n}(X_i-\bar X)^2$ be the sample
variance. In this note we give simple proofs of two well-known, but
related results:
\begin{enumerate}
\item[{\bf 1.1)}]
the probability density function of Student's t distribution with n
degrees of freedom converges to the standard normal density, and
\item[{\bf 1.2)}]
the sample standard deviation $S$ of a random sample from a normal
distribution with variance $\sigma^2$ is asymptotically unbiased for
$\sigma$.
\end{enumerate}
When taking a random sample of size n from a (nondegenerate) normal
distribution with mean $\mu$ and finite variance $\sigma^2$,
$(n-1)S^2/\sigma^2$ has the Chisquare distribution with $v=n-1$
degrees of freedom (d.f.) and probability density function (pdf)
\[f(x;v)=\frac{1}{2^{v/2}{\Gamma(\frac{v}{2})}}x^{v/2}e^{-x/2}\mbox{
for $x>0$}\] and the random variable $\sqrt{n}(\bar X -\mu)/S$ has
Student's t distribution with $v=n-1$ d.f. and pdf (e.g. see
\cite[page 230]{R})
\[f(t;v)=\frac{c_v}{\sqrt{2\pi}(1+t^2/v)^{(v+2)/2}},\;\;-\infty<t<\infty\]
\noindent where $c_v=\frac{\sqrt{2}\Gamma(\frac{v+1}{2})}
{\sqrt{v}\Gamma(\frac{v}{2})}$ and $\Gamma$ is the gamma function
defined on $r>0$ by $\Gamma(r)=\int_{0}^{\infty}x^{r-1}e^{-x}dx$.\\
Since $\lim_{v \rightarrow \infty} ((1+t^2/v)^{v})^{-1/2} =e^{-t^2/2}$
and $\lim_{n \rightarrow \infty}(1+t^2/n)^{-1/2}=1$, {\bf 1.1)} is
true if $lim_{v \rightarrow \infty}c_v=1$. Let $Y=(n-1)S^2/\sigma^2$.
Then, it is easy to see that $\frac{\sqrt{n-1}}{\sigma}E(S)=
E(Y^{1/2})=\sqrt{2}\Gamma(n/2)/\Gamma((n-1)/2$, i.e.
$E(S)=c_v\sigma$, where $v=n-1$ and so {\bf 1.2)} is also true if
$lim_{v\rightarrow \infty}c_v=1$.\\
Thus a simple proof of both {\bf 1.1)} and {\bf 1.2)} depends
on showing that $c_v$ tends to 1 as $v$ tends to $\infty$. This
crucial step is usually proved by invoking Stirling's approximation to
the $\Gamma$ function. Our proof does not use Stirling's approximation
to the gamma function, but uses the recursive property of the gamma
function, $\Gamma(x+1)=x\Gamma(x)$, elementary properties of
convergent sequences, and the fact that $E(S) \leq \sigma$.