How to prove $\lim\limits_{x\to \infty}3x=\infty$?
First I am not sure about formal definition of $\lim\limits_{x\to \infty}f(x)=\infty$, I guess $\forall K\in \Bbb{R},\exists N\in \Bbb{R}:x\gt N\implies f(x)\gt K$
If that's the case,
Let $K\in \Bbb{R}$, let $N=\frac{K}{3}$, then $x\gt \frac{K}{3}\implies 3x\gt K$.
I am sure it's not this simple. Could someone givea valid one?