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**Aryth** First it states to give an alternate proof of a theorem that showed that a sequence's limit is unique.

Choose $\displaystyle \epsilon > 0$. There is an $\displaystyle N_1$ such that for $\displaystyle n \geq N_1$, $\displaystyle |a_n - A| < \frac{\epsilon}{2}$ and there is an $\displaystyle N_2$ such that for $\displaystyle n \geq N_2$, $\displaystyle |a_n - B| < \frac{\epsilon}{2}$. Use the triangle inequality to show that this implies that $\displaystyle |A-B| < \epsilon$. Argue that $\displaystyle A = B$.

Now, I've already shown that the above implies that $\displaystyle |A-B| < \epsilon$, but I'm having trouble seeing how I can argue that the limits are equal (even though I know they are).