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Mathematics · Advanced university

Sequence limits and epsilon proofs

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

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Prerequisites

Logic, inequalities and real sequences.

Study notes

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

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.

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

  1. Which statement reverses the quantifiers incorrectly for convergence?

    1. There exists one N that works for every ε>0
    2. All sufficiently late terms lie within ε
    3. The cutoff may depend on ε
    4. For each ε>0 there is a suitable N

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

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

  4. What extra property makes every Cauchy sequence converge in R?

    1. Completeness
    2. Countability
    3. A chosen decimal display precision
    4. All terms being integers

Answers and marking guidance

Answers and marking notes
  1. There exists one N that works for every ε>0

    The needed cutoff generally grows as ε shrinks.

  2. 101

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

  3. 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.

  4. Completeness

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

Further reading and review references

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