Frequency Increases when Stiffness Increases

Problem Definition
Then as the force constant (or "stiffness") of $k$ increases, the frequency of the motion of $C$ likewise increases.

Proof
From Position of Cart attached to Wall by Spring, the horizontal position of $C$ is given as:
 * $(1): \quad x = C_1 \cos \alpha t + C_2 \sin \alpha t$

where:
 * $C_1$ and $C_2$ depend upon the conditions of $C$ at time $t = 0$
 * $\alpha = \sqrt {\dfrac k m}$

From the definition of frequency:

From Square Root is Strictly Increasing, the square root is strictly increasing on positive real numbers.

Hence, if the mass is constant (as is implicit in the statement of the problem), increasing $k$ causes an increase in $\nu$.