Proposition : A convergent sequence has a unique limit.

Proof : Suppose that (sₙ) converges to l and also to l’ where l’ ≠ l.
Put ε = 1/2 |l − l’|. Since (sₙ) converges to l, there is an integer N₁
such that |sₙ − l| < ε for all n ≥ N₁. Similarly, since (sₙ) converges to l’,
there is an integer N₂ such that |sₙ − l’| < ε for all n ≥ N₂ .

Put N = max{N₁, N₂}. Then, using the triangle inequality (Proposition 4.9),
|l − l’| = |l − sₙ + sₙ − l’ | ≤ |l − sₙ | + |sₙ − l’| < 2ε = |l − l’|.

This contradiction shows that l ≠ = l. [A proof by contradiction]