Let $U_{1}, ...,U_{i}$ independent random variables $U[0,1]$. Define
$$ I_{1} = \left\{ \begin{array}{l l} 1\ & \quad \text{if $U_{1}$ < $\frac{k}{n}$}\\ 0 & \quad \text{otherwize} \end{array} \right.$$ and
$$ I_{i} = \left\{ \begin{array}{l l} 1\ & \quad \text{if $U_{i+1}$ < $\frac{k-(I_{1}+...+I_{i})}{n-i}$}\\ 0 & \quad \text{otherwize} \end{array} \right.$$
demonstrate with: $$P[I_{1}=1] = \frac{k}{n}$$ and $$P[I_{i+1}=1|I_{1},...,I_{i}]=\frac{k-\sum_{j=1}^{i}I_{j}}{n-i},1<i<n$$
I have shown the first part ($P[I_{1}=1]=\frac{k}{n}$), just could not take the step of induction to demonstrate the rest
Obs: The k be the size of the sample, $k< n$