Mathematics · Advanced university
Sequence limits and epsilon proofs · revision notes
For every ε>0 there exists N such that every n≥N satisfies the error bound.
Key ideas and conditions
Quantifier order
For every ε>0 there exists N such that every n≥N satisfies the error bound.
Construct N
Solve the desired inequality for n, then choose an integer cutoff meeting it.
Cauchy criterion
A real sequence converges exactly when it is Cauchy, using completeness of the reals.
Counterexamples
One carefully verified counterexample refutes a universal claim; many examples do not prove it.
Explain without looking
- Prove a real sequence cannot converge to two different limits L and M.
- Does checking the first million terms establish convergence of every sequence?
Further reading and review references
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