Use the definition of a limit to prove that $\lim_{n\to \infty} 4^n = \infty$.

Question

Is my strategy valid?

I assume that $\exists M \in \mathbb{N}$ such that $\forall n$, $|4^n| \le M$, then I can take $a_M$ so $|a_M|=|4^M|>M$ for some $M$. Where this contradicts the assumptions, as such there is no such $M$. Is this approach OK?

Thanks.

Answer 1

You make it too convoluted, in my opinion; there is not need to look for a contradiction. Here how I would do it: given $M>0$, there exists $N>0$ such that $4^n>M$ for all $n>M$ (you can actually take $N=M$). That's exactly the definition of $\lim_{n\to\infty}4^n=\infty$.