Infinite Venn diagram

I think I had seen exactly this particular example in one of my real analysis books a few months ago, although I'm not sure about it. Anyway, here's a try at a proof:

Denote by $\{A_n\}_{n\geq 1}$ the class of open intervals where $A_n$ is defined as $A_n:=\left(0,\dfrac 1n\right)~\forall~n\in\Bbb{Z^+}$ .

Lemma: $\forall~m,n\in\Bbb{Z^+}~,~m\lt n\iff A_n\subset A_m$

Proof. $\forall~m,n\in\Bbb{Z^+}~,~0\lt m\lt n\iff \frac 1m\gt \frac1n\gt 0$

Hence, we can write $A_m=A_n\cup\left[\dfrac 1n,\dfrac 1m\right)\supset A_n\iff A_n\subset A_m$ .

Using the above lemma, we have,

$\begin{aligned}A_1\supset A_2\supset\cdots\supset A_m~\forall~m\in\Bbb{Z^+}&\implies \bigcap_{n=1}^m A_n = A_m\neq\emptyset~\forall~m\in\Bbb{Z^+}\\&\implies \bigcap_{n=1}^\infty A_n=\lim_{m\to\infty}\bigcap_{n=1}^m A_n=\lim_{m\to\infty} A_m\\&\implies \bigcap_{n=1}^\infty A_n=\lim_{m\to\infty}\left(0,\frac 1m\right)=(0,0)=\emptyset\end{aligned}$

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Does this count as a rigorous proof?