I was told that the half-normal distribution, with density: $ f(x)=\frac{1}{\sigma}\sqrt{\frac{2}{\pi}}\exp \left(-\frac{x^2}{2 \sigma^2} \right) \qquad \forall x \geq 0 $ and the exponential distribution, with parameter $\lambda>0$ and density : $ g(x) = \lambda \exp ( -\lambda x ) \qquad \forall x \geq 0 $ cannot be parametrised (ie, for any $\lambda>0$, and $\sigma>0$) so that a constant $c > 0$ exists such that $g(x) \leq c f(x) \qquad \forall x \geq 0$
Is this true ? Is there a proof for this, especially one that I am likely to be able to understand (high school mathematics) ? If not, is there a counter-example ?