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Mathematics · lesson 1 of 3

Sequence limits and epsilon proofs

For every ε>0 there exists N such that every n≥N satisfies the error bound.

Advanced university8 practice questionsNotes + worked examples
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Learn the ideas

Prerequisites: Logic, inequalities and real sequences.

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.

Worked examples

Example 1

Prove 1/n tends to zero using the epsilon definition.

Credit the order of quantifiers, an explicit N, and a bound for every n≥N.

Given ε>0, choose an integer N>1/ε. For n≥N, 0<1/n≤1/N<ε.

Example 2

Prove every convergent real sequence is bounded.

Credit both tail control and the finite initial segment.

If a_n→L, eventually abs(a_n−L)<1, so the tail is bounded by abs(L)+1. Take the maximum of that bound and the finitely many initial absolute values.

Check your understanding

8 questions

Decimals or fractions are accepted for numeric answers. Proofs and written responses use sample answers and require human marking. This is practice, not a protected examination.

1 Prove 1/n tends to zero using the epsilon definition.

Sample answer and marking guidance

Given ε>0, choose an integer N>1/ε. For n≥N, 0<1/n≤1/N<ε.

Credit the order of quantifiers, an explicit N, and a bound for every n≥N.

2 Prove every convergent real sequence is bounded.

Sample answer and marking guidance

If a_n→L, eventually abs(a_n−L)<1, so the tail is bounded by abs(L)+1. Take the maximum of that bound and the finitely many initial absolute values.

Credit both tail control and the finite initial segment.

3 Which statement reverses the quantifiers incorrectly for convergence?

Show answer and explanation

There exists one N that works for every ε>0

The needed cutoff generally grows as ε shrinks.

4 What is the smallest positive integer N such that 1/n<0.01 for every n≥N?

Show answer and explanation

101

The strict inequality requires n>100, not n≥100.

5 Give a bounded real sequence that does not converge and justify it.

Sample answer and marking guidance

a_n=(−1)^n is bounded by 1 in absolute value, but its even and odd subsequences have limits 1 and −1.

Credit boundedness and a valid nonconvergence argument.

6 What extra property makes every Cauchy sequence converge in R?

Show answer and explanation

Completeness

Completeness prevents a Cauchy sequence from approaching a missing real-number limit.

7 Prove a real sequence cannot converge to two different limits L and M.

Sample answer and marking guidance

Assume L≠M and choose ε=abs(L−M)/3. Far enough out, the triangle inequality would give abs(L−M)<2ε, a contradiction.

Credit one common tail index and a contradiction via the triangle inequality.

8 Does checking the first million terms establish convergence of every sequence?

Show answer and explanation

No

Convergence is a statement about all sufficiently large indices, not a finite prefix.

Further reading and review references

These are reference links, not copied exercises, institutional endorsement or proof of independent review.