The pdf of the order statistic can be written as
\[g(y_1,y_2,...,y_n)=\begin{cases} n!f(y_1)f(y_2)...f(y_n) & a<y_1<y_2<...y_n\\ 0 &\text{elsewhere} \end{cases}\]
The marginal pdf of any order statistic, say \(Y_k\) can be calculated by the following formula
\[\begin{align} g_k(y_k)&=\int_a^{y_2}\int_a^{y_3}...\int_a^{y_k}\int_{y_k}^b\int_{y_{k+1}}^b...\int_{y_{n-1}}^b n!f(y_1)f(y_2)...f(y_{k-1})f(y_k)f(y_{k+1})...f(y_n)dy_n...dy_{k-1}dy_{k+1}...dy_2dy_1\\ &=n!f(y_k)\int_a^{y_2}f(y_1)dy_1... \int_a^{y_k}\int_{y_k}^b\int_{y_{k+1}}^b...\int_{y_{n-1}}^b n!f(y_1)f(y_2)f(y_n)dy_n...dy_{k-1}dy_{k+1}...dy_2\\ &=n!f(y_k)\int_a^{y_3}F(y_2)f(y_2)dy_2... \int_a^{y_k}\int_{y_k}^b\int_{y_{k+1}}^b...\int_{y_{n-1}}^b n!f(y_1)f(y_2)f(y_n)dy_n...dy_{k-1}dy_{k+1}...dy_3\\ &=...\\ &=\frac{n!}{(k-1)!(n-k)!}[F(y_k)]^{k-1}[1-F(y_k)]^{n-k}f(y_k) \tag{1} \end{align}\]
For above calculations we need to use the following calculations:
\[\int_a^{y_2} f(y_1)dy_1=\int_a^{y_2}dF(y_1）=F(y_2)-F(a)=F(y_2) \text{, since } F(a)=0\] \[\int_a^{y_3}F(y_2)f(y_2)dy_2=\int_a^{y_3}F(y_2)dF(y_2)=\frac{1}{2}F(y_3)^2\] We continue this process until \(f(y_{k-1})\) and we get \(\frac{1}{(k-1)!}[F(y_k)]^{k-1} \tag{2}\)
(2)For \(y_{k+1}, y_{k+2},...y_n\) we have the following calculations:
\[\int_{y_{n-1}}^bf(y_n)dy_{n}=\int_{y_{n-1}}^bdF(y_{n})=F(b)-F(y_{n-1})=1-F(y_{n-1}) \text{, since } F(b)=1\] \[\int_{y_{n-2}}^b [1-F(y_{n-1})]f(y_{n-1})dy_{n-1}=\int_{y_{n-2}}^b [1-F(y_{n-1})]dF(y_{n-1})=\frac{1}{2}[1-F(y_{n-2})]^2\] We continue this process until \(f(y_{k+1})\) and we get \(\frac{1}{(n-k)!}[1-F(y_k)]^{n-k} \tag{3}\)
Combine \((2)\) and \((3)\) we can get \((1) \blacksquare\)