Uniqueness of limits

The idea of the proof is that two distinct numbers \(L\) and \(M\) can be separated by disjoint neighborhoods. Let \(\varepsilon=\frac{|L-M|}{2}>0\). If \(a_n\to L\), there is \(N_1\) with \(|a_n-L|<\varepsilon\) for \(n\ge N_1\); if \(a_n\to M\), there is \(N_2\) with \(|a_n-M|<\varepsilon\) for \(n\ge N_2\). Taking \(n\ge\max(N_1,N_2)\) both hold at once.

Then the triangle inequality gives \(|L-M|\le|L-a_n|+|a_n-M|<2\varepsilon=|L-M|\), that is, \(|L-M|<|L-M|\), absurd. Hence \(L=M\). An equivalent variant shows \(|L-M|<\varepsilon\) for every \(\varepsilon>0\), which forces \(|L-M|=0\).

Uniqueness has a practical use: to show that a sequence does not converge it suffices to see that, if it did, it would have to meet two incompatible conditions. For \((-1)^n\), a limit \(L\) would have to be within \(\tfrac12\) of both \(1\) and \(-1\), which is impossible because they are at distance \(2\).

The result holds in \(\mathbb R\) thanks to the absolute value and the triangle inequality. In spaces with other notions of closeness, two distinct points must also be separable, which leads to the idea of a Hausdorff space.