Convergence of inverse of a null sequence

Be $(a_n) \subset \mathbb{R}$ a null sequence, that is, $\lim_{n \to \infty} a_n = 0$. We want to prove that the sequence $(\frac{1}{a_n})$ is not convergent.

As we know that $(a_n)$ is a null sequence, we know that for all $\epsilon > 0$ there exists an $n_0 \in \mathbb{N}$ such that for all $n \in \mathbb{N}$ with $n \geq n_0$ we have that $|a_n| < \epsilon$.

Now, let’s consider the sequence $(\frac{1}{a_n})$, (suppose that $a_n \neq 0$ for all $n \in \mathbb{N}$) we know that $|a_n| < \epsilon$, then we can say that $\frac{1}{|a_n|} > \frac{1}{\epsilon}$. So for each $\epsilon \in \mathbb{R}, \epsilon > 0$, we can find an $n_1 \in \mathbb{N}$ such that for all $n \in \mathbb{N}$ with $n \geq n_1$ we have that $\frac{1}{|a_n|} > \frac{1}{\epsilon}$. Therefore, $(1/a_n)$ is not bounded, and so it is not convergent.