Math 2.0

Real functions Proof

Proof of the sequential characterization of limits

Builds on Limit of a functionEpsilon-delta definition of a limitEpsilon-N definition of the limit of a sequenceThere is a natural number n with 1/n below any positive epsilonLimit of a sequence

Proof

We use the epsilon-delta form of the limit and the epsilon-N form for sequences. The converse is proved by contraposition.

  1. Let D⊆ℝD \subseteq \R, let f ⁣:D→ℝf \colon D \to \R, let aa be a limit point of DD, and let L∈ℝL \in \R.
  2. Assume first that lim⁡x→af(x)=L\lim_{x \to a} f(x) = L, in the sense of the limit of a function. Let (xn)(x_n) be a sequence in D∖{a}D \setminus \set{a} with xn→ax_n \to a. Let ε>0\varepsilon > 0.
  3. The epsilon-delta form gives δ>0\delta > 0 with ∣f(x)−L∣<ε\abs{f(x) - L} < \varepsilon for all x∈Dx \in D with 0<∣x−a∣<δ0 < \abs{x - a} < \delta.
  4. Since xn→ax_n \to a, the epsilon-N form gives NN with ∣xn−a∣<δ\abs{x_n - a} < \delta for all n≥Nn \ge N.
  5. For n≥Nn \ge N, also xn≠ax_n \ne a, so 0<∣xn−a∣<δ0 < \abs{x_n - a} < \delta. Step 3 gives ∣f(xn)−L∣<ε\abs{f(x_n) - L} < \varepsilon.
  6. So f(xn)→Lf(x_n) \to L by the epsilon-N form. This proves the first direction.
  7. Conversely, assume that lim⁡x→af(x)=L\lim_{x \to a} f(x) = L fails. We show that the sequence condition fails.
  8. By the epsilon-delta form, there is ε>0\varepsilon > 0 with this property: for each δ>0\delta > 0 there is x∈Dx \in D with 0<∣x−a∣<δ0 < \abs{x - a} < \delta and ∣f(x)−L∣≥ε\abs{f(x) - L} \ge \varepsilon.
  9. For each natural number nn, apply step 8 with δ=1/(n+1)\delta = 1/(n+1). This gives xn∈Dx_n \in D with 0<∣xn−a∣<1/(n+1)0 < \abs{x_n - a} < 1/(n+1) and ∣f(xn)−L∣≥ε\abs{f(x_n) - L} \ge \varepsilon.
  10. Each xnx_n lies in D∖{a}D \setminus \set{a}, since xn≠ax_n \ne a.
  11. Let η>0\eta > 0. There is m≥1m \ge 1 with 1/m<η1/m < \eta. For n≥mn \ge m, ∣xn−a∣<1/(n+1)<1/m<η\abs{x_n - a} < 1/(n+1) < 1/m < \eta. So xn→ax_n \to a.
  12. But ∣f(xn)−L∣≥ε\abs{f(x_n) - L} \ge \varepsilon for every nn. So no NN works for this ε\varepsilon in the epsilon-N form, and f(xn)↛Lf(x_n) \not\to L.
  13. So the sequence condition fails. By contraposition, the sequence condition implies lim⁡x→af(x)=L\lim_{x \to a} f(x) = L.

Idea

If f(x)f(x) is close to LL near aa, then f(xn)f(x_n) is close to LL once xnx_n is near aa. If the limit fails, one ε\varepsilon is violated arbitrarily close to aa. Picking one bad point within 1/(n+1)1/(n+1) of aa for each nn gives a sequence that witnesses the failure.

Reading path

To understand this page from the ground up, read these 50 pages in order. The foundations come first. The path takes the proofs Proof of the approximation property of the supremum, Proof that there is a natural number n with 1/n below any positive epsilon, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the epsilon-N definition of the limit of a sequence, Proof of the epsilon-delta form of the limit of a function, Proof that the reals are Archimedean from the completeness axiom.

  1. Construction of the reals from Cauchy sequences concept
  2. Element definition
  3. Integers object
  4. Limit point of a set definition
  5. Ordered field definition
  6. Proposition concept
  7. Rational numbers object
  8. Real numbers object
  9. Sequence definition
  10. Set bounded above definition
  11. Set definition
  12. Ring of integers statement
  13. Limit of a function definition
  14. Archimedean ordered field definition
  15. Implication definition
  16. Predicate definition
  17. The reals form a field statement
  18. Real sequence definition
  19. Supremum definition
  20. Ordered pair definition
  21. The rationals form a field statement
  22. Universal quantifier definition
  23. Limit of a sequence definition
  24. Completeness axiom statement
  25. Cartesian product definition
  26. Subset definition
  27. The rationals form an ordered field statement
  28. Binary relation definition
  29. Order on the reals definition
  30. Antisymmetric relation definition
  31. Reflexive relation definition
  32. Transitive relation definition
  33. Interval definition
  34. Preorder definition
  35. Proof of the epsilon-delta form of the limit of a function proof
  36. Partial order definition
  37. Epsilon-delta definition of a limit statement
  38. Total order definition
  39. Proof that the reals form an ordered field, with the Cauchy sequence construction proof
  40. The reals form an ordered field statement
  41. Absolute value definition
  42. Proof of the approximation property of the supremum proof
  43. Proof of the epsilon-N definition of the limit of a sequence proof
  44. Approximation property of the supremum statement
  45. Epsilon-N definition of the limit of a sequence statement
  46. Proof that the reals are Archimedean from the completeness axiom proof
  47. The reals are Archimedean statement
  48. Proof that there is a natural number n with 1/n below any positive epsilon proof
  49. There is a natural number n with 1/n below any positive epsilon statement
  50. Proof of the sequential characterization of limits proof

Where this leads

1111 pages build on this one.

Further reading

  1. Web https://www.jirka.org/ra/, Lemma 3.1.7
About this page
TagGK54EWGGR4
URIm2://GK54EWGGR4
Kindproof
WritingW0 stub
MSC26A03
ProvesSequential characterization of limits
UsesLimit of a function, Epsilon-delta definition of a limit, Epsilon-N definition of the limit of a sequence, There is a natural number n with 1/n below any positive epsilon, Limit of a sequence
Changed4 November 2026

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