The two properties are not "contradictory", they are complementary. Both of them are true.
(To say that they are contradictory would be like saying that "$30 = 2\times 15$" is contradictory with "$30 = 3\times 10$". They aren't contradictory, they can both hold at the same time).
For inner products over the real numbers, both equalities hold: $\langle c\mathbf{u},\mathbf{v}\rangle = c\langle\mathbf{u},\mathbf{v}\rangle = \langle\mathbf{u},c\mathbf{v}\rangle$ for all vectors $\mathbf{u}$ and $\mathbf{v}$ and all scalars $c$.
In order to prove it, however, one needs to know exactly what properties of the inner product you are assuming. I'm guessing that they are the following:
- $\langle \mathbf{u},\mathbf{u}\rangle\geq 0$ for all $\mathbf{u}$; $\langle \mathbf{u},\mathbf{u}\rangle = 0$ if and only if $\mathbf{u}=\mathbf{0}$;
- $\langle \mathbf{u}+\mathbf{w},\mathbf{v}\rangle = \langle \mathbf{u},\mathbf{v}\rangle + \langle\mathbf{w},\mathbf{v}\rangle$ for all $\mathbf{u},\mathbf{v},\mathbf{w}$.
- $c\langle \mathbf{u},\mathbf{v}\rangle = \langle c\mathbf{u},\mathbf{v}\rangle$ for all $\mathbf{u}, \mathbf{v}$ and all $c$.
- $\langle\mathbf{u},\mathbf{v}\rangle = \langle\mathbf{v},\mathbf{u}\rangle$ for all $\mathbf{u},\mathbf{v}$.
If this ist he case, start with $\langle \mathbf{u},c\mathbf{v}\rangle$, and then use 4, 3, and 4 again to get the desired result.