#### How do I prove this summation question over real a number?

I need to prove the following:

\sum_{k=0}^{n\ -1}\ \ \lfloor\ x+\frac{k}{n}\ \rfloor\ =\ \ \lfloor nx\rfloor

I can see how to prove this if x is an integer.

Since we know that each \frac{k}{n}<\ 1, \ \lfloor x\ +\frac{k}{n}\rfloor\ =\ x since x is an integer and x + k/n < x + 1. The LHS becomes nx.

Since x is an integer, the RHS is also nx. Proving the same when x is an integer.

How do I prove this when x is a real number?