In the book "A graduate course in probability" by Allan Gut, I ran into the following lemma. I have no issues with this, however, in the very next page, Gut states that by applying this lemma along with the function $g(x)=e^{tx}$, we obtain $$\mathbb{P}(X>x)\leq \frac{\mathbb{E}[e^{tX}]}{e^{tx}},$$
Here, $X$ is a r.v. with mean $0$ and variance $\sigma^2$. Also $\mathbb{P}(|X|\leq b)=1$ for some real $b$ and $0<t<b^{-1}$. I don't have a problem believing the statement, I can even prove it by writing $\mathbb{P}(X>x)=\mathbb{P}(e^{tX}>e^{tx})$, and then continue with the said lemma, but THAT is not what Gut says he does. Any insight would be appreciated. Thank you