Primary Tool
Proof Builder
This layout keeps the contradiction argument readable from start to finish. You can see the choice of , the two index bounds, and the final inequality as separate moves instead of one compressed paragraph.
What the theorem says
Suppose a sequence 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 converges to both and with . The distance is then positive, so taking gives a concrete amount of separation.
Past a large enough index, the sequence must lie within of and within of at the same time. Triangle inequality then says is smaller than , 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?
Related Pages
Keep moving through the cluster
Proof that every convergent sequence is bounded
This theorem is a standard follow-up to the definition of convergence because it shows how one tail estimate plus finitely many early terms gives a global bound on the whole sequence.
Open proof →
What convergence means for sequences
A sequence converges when its terms eventually stay as close as you want to a single number. The word eventually matters more than the first few terms.
Open explanation →
What an epsilon-delta proof is actually controlling
An epsilon-delta proof is a control problem: keep close enough to a point so the function value stays inside a target band around the limit.
Open explanation →