Bunching up without a target

Mathematics · Calculus & Analysis · ages 20-21

Name ______________________ Date ____________

For a_n = 1/n, find N so that n > N forces a_n < 0.01.
Answer: ______________
A sequence is Cauchy when its terms do what?
- stay bounded
- converge to zero
- eventually bunch together
- are all rational
A sequence is Cauchy when its terms do what?
- Stay bounded
- Converge to zero
- Eventually bunch together
Every convergent sequence is Cauchy.
Circle one: True False
For a_n = 1/n, what is a_100 as a decimal?
Answer: ______________
Mia says Bolzano-Weierstrass means every bounded sequence converges. Is Mia right?
Circle one: True False
The sequence 1, 1.4, 1.41, 1.414, ... is Cauchy in the rationals with what property?
- no rational limit
- rational limit 1.5
- rational limit 2
- no real limit
Showing terms get pairwise close without naming a limit proves a sequence is Cauchy. True or false?
Circle one: True False
From the bounded divergent sequence (-1) to the n, which subsequence converges?
- The whole sequence
- The even indexed terms, all equal to 1
- Every third term
From the bounded divergent sequence (-1)^n, which subsequence converges?
- the whole sequence
- terms 1, 2, 3 in order
- every third term
- the even indexed terms, all equal to 1

Answer key

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

100 · 100. Since 1/n < 0.01 exactly when n > 100.
eventually bunch together · Eventually bunch together. Past some index, all pairs sit within any prescribed tolerance.
Eventually bunch together · Cauchy means late terms sit pairwise close, with no limit named.
True · Terms near the limit are near each other by the triangle inequality.
0.01 · 0.01. Note 1/100 = 0.01.
False · The theorem promises a convergent subsequence, not convergence of the whole sequence.
no rational limit · No rational limit. The decimals chase root 2, which is missing from Q.
True · True. Pairwise bunching is the definition; limits enter only afterward via completeness.
The even indexed terms, all equal to 1 · Constant subsequences are the first extraction drill.
the even indexed terms, all equal to 1 · The even indexed terms, all equal to 1. A constant subsequence converges trivially.