Proof that bases can be extended while maintaining orthonormality

Question

If we have a subspace $U$ of $V$, where $U$ has an orthonormal basis $(u_1, u_2, \dots, u_n)$, how do we show that we can extend the basis to $(u_1,u_2,\dots,u_n,v_1,v_2,\dots,v_r)$ such that it is orthonormal? I get the extending the basis part, but how do we also prove that it can be orthonormal? That is the part I am stuck at.