Let $m$ be a measure. We assume that there exists a constant $C > 0$ such that inequality (3)
$$\int e ^ { \lambda f ( x ) } m ( x ) d x \leqslant \exp \left( \lambda \int f ( x ) m ( x ) d x + \frac { C \lambda ^ { 2 } } { 4 } \right)$$
applies to all $f \in \mathscr{C}_b^1$ with $|f'(x)| \leq 1$. Show that inequality (3) applies to the function defined by $f ( x ) = x$. You may use the sequence of functions defined by $f _ { n } ( x ) = n \arctan \left( \frac { x } { n } \right)$.