Cauchy Sequences and the Bolzano-Weierstrass Theorem · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

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Mathematics · Calculus & Analysis · ages 20-21
Name ______________________   Date ____________

  1. For a_n equal to 2 over n, what is the smallest integer N such that every n greater than or equal to N gives a_n below 0.03? Give N.

    Answer: ______________

  2. For a_n equal to 5 plus 1 over n, which statement best explains why it is Cauchy?

    • It stays bounded, even if late terms can stay pairwise far apart.
    • It eventually becomes exactly 5.
    • Late pairwise differences of 1 over n become small.
    • It alternates between two fixed values.

  3. A sequence is Cauchy when its terms do what?

    • Stay bounded inside one fixed interval forever, even if late terms still stay pairwise far apart and no bunching claim follows
    • Converge to zero, regardless of their starting values and formula
    • Eventually bunch together within every tolerance, without naming a limit

  4. Every convergent sequence is Cauchy.

    Circle one:   True   False

  5. Mia says Bolzano-Weierstrass means every bounded sequence converges. Is Mia right?

    Circle one:   True   False

  6. In the bisection proof of Bolzano-Weierstrass, what is kept after each split?

    • A half interval containing infinitely many sequence terms
    • The half with the largest endpoint, even if it has finitely many terms
    • Both halves, because a subsequence must use every term
    • Only the interval containing the first term

  7. What follows for a Cauchy sequence of real numbers?

    • It must be eventually constant.
    • It converges to some real limit.
    • It must alternate between two values.
    • It can have no bounded tail.

  8. The sequence 1, 1.7, 1.73, 1.732, and so on is Cauchy in the rationals with what property?

    • No rational limit
    • Rational limit 1.7
    • Rational limit 3
    • No real limit

  9. From the bounded divergent sequence 3, 0, 3, 0, and so on, which subsequence converges?

    • The full sequence, because it is bounded
    • The odd indexed terms, all equal to 3
    • The paired blocks 3, 0, 3, 0, and so on, which still diverge

  10. For a_n equal to 2 over n, what is the smallest integer N such that every m and n greater than or equal to N satisfy the distance between a_m and a_n below 0.2? Give N.

    Answer: ______________

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Answer key

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  1. 67 · Two over n below 0.03 means n is greater than 66 and two thirds, so 67 is the first valid cutoff.
  2. Late pairwise differences of 1 over n become small. · Subtracting the constant 5 leaves differences of 1 over n, and those late differences become arbitrarily small.
  3. Eventually bunch together within every tolerance, without naming a limit · Cauchy means late terms sit pairwise close, with no limit named.
  4. True · Terms near the limit are near each other by the triangle inequality.
  5. False · The theorem promises a convergent subsequence, not convergence of the whole sequence.
  6. A half interval containing infinitely many sequence terms · Keep a half interval containing infinitely many terms. That invariant lets the extraction continue forever.
  7. It converges to some real limit. · Completeness of the real numbers turns every real Cauchy sequence into a convergent sequence.
  8. No rational limit · It chases root 3, which is missing from the rationals.
  9. The odd indexed terms, all equal to 3 · The odd indexed terms are constant at 3, so that subsequence converges.
  10. 10 · For finite m,n at least 10, the difference is strictly less than 2 over m, which is at most 0.2. N equal to 9 fails with m equal to 9 and n equal to 90, where the difference is exactly 0.2.
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