Why Any $B$-bounded centered random variable $X$ is subgaussian with parameter $B \sqrt{2\pi}$

Question

Definition of subgaussian is below

A random variable $X$ over $\mathbb{R}$ is subgaussian with parameter $\alpha > 0$ if its moment generating function satisfies $\mathbb{E}[exp(2 \pi tX)] \leq exp(\pi \alpha^2 t^2)$ for all $t \in \mathbb{R}$.

And there is a fact about subgaussian.

Any $B$-bounded centered random variable $X$ is subgaussian with parameter $B \sqrt{2\pi}$.

Why is the parameter $B \sqrt{2\pi}$? I cannot understand above fact.

Answer 1

Hint: prove first that for a mean zero random variable bounded by $B$, $$\mathbb{E}[exp(2 \pi tX)] \leq \mathbb{E}[exp(2 \pi tY)]$$ where $Y$ is a random variable with $\mathbb{P}(Y=B)=0.5$ and $\mathbb{P}(Y=-B)=0.5$. proving the theorem for $Y$ is much simpler!