If I'm doing a proof and I use vertical dots or specifically say something like "carrying on in this way until", is that considered a rigorous proof?
Here's a relatively simple example of what I mean from group theory. Let's say I want to prove that for every $a_1, a_2, \dots , a_n\in G$, $$(a_1a_2\cdots a_n)^{-1} = a_n^{-1}\cdots a_2^{-1}a_1^{-1}$$ Here's how I might do it:
Let $a_1,\dots, a_n\in G$ with $n$ some positive integer. Then to show $(a_1\cdots a_n)^{-1} = a_n^{-1}\cdots a_1^{-1}$, it suffices to show that $(a_1\cdots a_n)(a_n^{-1}\cdots a_1^{-1}) = e$. Using the generalized associative law, we see that $$\begin{align*} (a_1\cdots a_n)(a_n^{-1}\cdots a_1^{-1}) &= (a_1\cdots a_{n-1})(a_na_n^{-1})(a_{n-1}^{-1}\cdots a_1^{-1}) \\ &= (a_1\cdots a_{n-1})(a_{n-1}^{-1}\cdots a_1^{-1}) \\ &= (a_1\cdots a_{n-2})(a_{n-1}a_{n-1}^{-1})(a_{n-2}^{-1}\cdots a_1^{-1}) \\ &\ \ \ \ \ \vdots \\ &=a_1a_1^{-1} \\ &= e\end{align*}$$ concluding the proof.
To be considered rigorous should I explicitly use an induction proof, or is this considered an appropriate argument?
