Wave Number of Harmonic Wave/Proof 2

Proof
By definition, $k$ is the number of complete wavelengths of $\phi$ per unit distance along the $x$-axis.

By definition, $\lambda$ is the period of the wave profile of $\phi$.

So between two points at unit distance apart, there are $\dfrac 1 \lambda$ wavelengths of $\phi$.

The result follows by definition of wave number.