I have a proof of uniqueness of sequence limits I would like to check for holes in logic. It's a bit non-standard (meaning I see another idea of most of proofs for this theorm I see online), and thus I am not sure it's correct one.
Let's assume we have one sequence $(a_n)$ and we have two limits: $(a_n) \to a_1, (a_n) \to a_2$. Instead of assuming $a_1 \neq a_2$ and getting contradiction, I have to prove that $a_1 = a_2$ directly.
Let's take arbitrary $\epsilon \gt 0$. I want to show that $|a_1 - a_2| \lt \epsilon$; since $\epsilon$ is arbitrary, that would prove the equality. Let's take $\epsilon_1 = \epsilon_2 = \epsilon /2$. Due to definition of sequence limit, I have $N_1$ such that $\forall n \ge N_1: |a_1 - a_n| \lt \epsilon_1$. I have the similar statement with $N_2, a_2$ and $\epsilon_2$.
Now, if I take $N = \max\{N_1, N_2\}$, and using triangle inequality, I can write: $$\forall n \ge N: \epsilon = \epsilon_1 + \epsilon_2 \gt |a_1 - a_n| + |a_n - a_2| \ge |a_1 - a_n + a_n - a_2| = |a_1 - a_2|$$
As the comments suggest, you cannot use the same label for different objects, since they cannot be formally distinguished inside the logic system, even if you can distinguish them in your mind. Besides that, the proof is fine.