I have been trying a question from Van Handel's HDP book

Show that if $\mathbf{E}\left[e^{X^{2} / 6 C}\right] \leq 2$, then $X$ is $18 C$-subgaussian.

And How is this related to :If $X\_{1}, \ldots, X\_{n}$ are independent, $C$-subgaussian, with mean 0 , then $\frac{1}{\sqrt{n}} \sum\_{i=1}^{n} X\_{i}$ is $18 C$-subgaussian.

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It is given.

Hint: for large values of $\lambda$, use Young's inequality $|\lambda X| \leq \frac{a \lambda^{2}}{2}+\frac{X^{2}}{2 a}$ for a suitable choice of $a$; for small values of $\lambda$, use Young's inequality together with $\mathbf{E}\left[e^{\lambda X}\right] \leq 1+\frac{\lambda^{2}}{2} \mathbf{E}\left[X^{2} e^{|\lambda X|}\right]$ by Taylor's theorem.

But I am unable to complete this proof, Can anyone help me in this?