Proof that the limit of a convergent sequence is unique

Primary Tool

Proof Builder

This layout keeps the contradiction argument readable from start to finish. You can see the choice of $ε$ , the two index bounds, and the final inequality as separate moves instead of one compressed paragraph.

What the theorem says

Suppose a sequence $(a_{n})$ converges. The claim is that there is only one real number it can converge to.

This theorem matters because later definitions in analysis would be unstable if limits were not unique. The proof is one of the cleanest places to see how the formal limit definition forces a contradiction.

Why the contradiction works

Assume $(a_{n})$ converges to both $L$ and $M$ with $L \neq = M$ . The distance $∣ L - M ∣$ is then positive, so taking $ε = ∣ L - M ∣/3$ gives a concrete amount of separation.

Past a large enough index, the sequence must lie within $ε$ of $L$ and within $ε$ of $M$ at the same time. Triangle inequality then says $∣ L - M ∣$ is smaller than $2 ε = 2∣ L - M ∣/3$ , which is impossible.

Common Pitfall

The contradiction is not about notation. It is that the same tail of the sequence would have to stay close to two different numbers that are a fixed positive distance apart.

Try a Variation

Try rewriting the same proof for limits of functions at a point. Which steps stay the same, and which parts need neighborhoods instead of indices?

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