Let $\mathcal{C}^{1}$ be the space of functions of class $C^{1}$ from $[-\pi, \pi]$ to $\mathbf{C}$. For $f \in \mathcal{C}^{1}$, we set
$$\|f\|_{\infty} = \max\{|f(t)|; t \in [-\pi, \pi]\} \quad \text{and} \quad V(f) = \int_{-\pi}^{\pi} |f^{\prime}|.$$
By considering a well-chosen sequence of functions, show that there does not exist an element $C$ of $\mathbf{R}^{+*}$ such that
$$\forall f \in \mathcal{C}^{1}, \quad V(f) \leq C\|f\|_{\infty}$$