injective sequence of natural numbers has infinite plus limit

Question

Question

Injective sequence of natural numbers $(a_n)$ has $\lim_{n\to+\infty}a_n=+\infty$.

Draft

I thought, if it's injective, like $\mathbb N$ is not majored, it's not bounded, so it doesn't converge. Then, if it doesn't converge, one of the two, either the limit is plus infinite or it doesn't exist. But how do you prove it doesn't exist?

Answer 1

For any $R > 0$ there are only finitely many indices $n$ with $a_n \le R$, since $(a_n)$ is injective. So we can define $$ N(R) = \max \{ n \mid a_n \le R \} \, . $$

Then for all $R > 0$, $n > N(R) \implies a_n > R$, and that proves $\lim_{n \to \infty} a_n = +\infty$.