Understanding the proof for Properties of Subgaussian Variables

Question

Here are the definitions, statements and the proof that I am stuck on:

I am stuck on the last part of the proof where the author claims that setting $C = e$ automatically guaranties that (1) holds for $0 < t < eK$ as well. I imagine what he meant is that given we have shown (4) implies (1) for $t \geq eK$ with $C = 1$, we can prove (1) again by updating $C = e$. However, why does this follow? I think I might be missing something obvious here.

Answer 1

Suppose $0<t<eK$. Then, $P(|X|>t) \leq 1 \le (e) (e^{-vt^{2}})$ since $e^{vt^{2}}\le e^{ve^{2}K^{2}}=e$.