Primary Tool
Proof Builder
Proof Builder keeps the two-part structure visible: first bound the tail using convergence, then fold the finitely many earlier terms into one maximum. That split is the whole proof.
Why the proof starts with
Convergence says that eventually the sequence stays within any positive distance of its limit. To prove boundedness you do not need a tiny distance. You only need one concrete tail bound.
Choosing gives a simple statement: after some index , every term lies inside the interval . That immediately bounds the tail by .
Where the maximum comes from
Convergence never says anything directly about the first finitely many terms. Those terms could be large, but there are only finitely many of them.
That is why the proof takes a maximum of the finitely many head terms together with the tail bound . One number then controls the entire sequence.
Common Pitfall
The convergence estimate only controls terms after some index. You still need a separate argument for the finitely many earlier terms, which is why the proof introduces a maximum.
Try a Variation
Try rewriting the proof with instead of . Which part of the argument stays the same, and which constant changes?
Related Pages
Keep moving through the cluster
Proof that the limit of a convergent sequence is unique
A convergent sequence cannot settle down to two different numbers. The proof is short, but it is a good model for how contradiction and the definition of limit work together.
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 →