به انگلیسی

Bunching up without a target

Mathematics · Calculus & Analysis · ages 20-21

Name ______________________ Date ____________

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: ______________
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.
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
Every convergent sequence is Cauchy.
Circle one: True False
Mia says Bolzano-Weierstrass means every bounded sequence converges. Is Mia right?
Circle one: True False
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
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.
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
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
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: ______________

Answer key

For grown-ups. Fold this page away before handing over the rest.

Bunching up without a target W1-mt_nreDylVkSU-s1

67 · Two over n below 0.03 means n is greater than 66 and two thirds, so 67 is the first valid cutoff.
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.
Eventually bunch together within every tolerance, without naming a limit · Cauchy means late terms sit pairwise close, with no limit named.
True · Terms near the limit are near each other by the triangle inequality.
False · The theorem promises a convergent subsequence, not convergence of the whole sequence.
A half interval containing infinitely many sequence terms · Keep a half interval containing infinitely many terms. That invariant lets the extraction continue forever.
It converges to some real limit. · Completeness of the real numbers turns every real Cauchy sequence into a convergent sequence.
No rational limit · It chases root 3, which is missing from the rationals.
The odd indexed terms, all equal to 3 · The odd indexed terms are constant at 3, so that subsequence converges.
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.