Topic 4 - Sequences

Question 1

Define: Sequence

Answer 1

A function from the natural numbers to the real numbers (IN to IR).

Question 2

Define: Convergence

Answer 2

A real sequence (a_n) that tends to a real number as n tends to infinity.

Question 3

Define: Limit

Answer 3

L is the limit of a real sequence (a_n) if:

∀ϵ > 0, ∃N ∈ IN s.t. ∀n ≥ N, |(a_n) - L| < ϵ.

We say (a_n) → L as n→∞, or lim_{n→∞} (a_n) = L

Question 4

T/F: There are cases where a real sequence has more than one limit.

Answer 4

False. This is false according to the “uniqueness of limits” theorem, which can be proved by contradiction.

Question 5

Definition: Bounded Sequence

Answer 5

A sequence (a_n) is bounded if the set of all its members is bounded.

Question 6

T/F: If a sequence converges to a limit, then the sequence is bounded.

Answer 6

True. This can be proved by using the definition of limits (see section 4.1 for full proof).

Question 7

Definition: Divergence

Answer 7

(a_n) diverges to ∞ if:

∀M ∈ R, ∃N ∈ IN s.t. ∀n ≥ N, (a_n) > M.

We say (a_n) → ∞ as n→∞, or lim{n→∞}(a_n) = ∞.

Similarly, (a_n) diverges to -∞ if ∀m ∈ R, ∃N ∈ IN s.t. ∀n ≥ N, (a_n) < m.

Question 8

Define: Monotone

Answer 8

A real sequence (a_n) is monotone if it is:

• increasing: (a_n) ≤ (a_{n+1}),
• strictly increasing: (a_n) < (a_{n+1}),
• decreasing: (a_n) ≥ (a_{n+1}),
• strictly decreasing: (a_n) > (a_{n+1}),

for all n ∈ IN.

Question 9

T/F: EVERY real sequence which is monotone and bounded converges.

Answer 9

True. This can be proved using the completeness axiom, Archimedean Postulate, and definition of limits.

Question 10

Define: Subsequence

Answer 10

Given a real sequence (a_n) and a strictly increasing sequence of natural numbers (n_k), then (a_{n_k}) is a subsequence of (a_n).

Question 11

T/F: If a subsequence converges to L, then the sequence converges to L.

Answer 11

False. Take cos(nπ/2) as an example. The subsequence of even numbers converges to 0, but the sequence doesn’t converge as it’s periodic.

The reverse implication of this statement is true however (see section 4.4).

Question 12

State: Bolzano-Weirstrass Theorem

Answer 12

Every bounded real sequence has a convergent subsequence.

Question 13

Definition: Cauchy Sequence

Answer 13

(a_n) is a cauchy sequence if

∀ϵ > 0, ∃N ∈ IN s.t. ∀m,n ≥ N,|a_m−a_n|< ϵ

(The difference between 2 terms of the sequence is smaller than any nonzero, positive number.)

Question 14

T/F: A sequence is cauchy, if and only if the sequence is convergent.

Answer 14

True. Both implications can be proved using the triangle inequality (see section 4.5)

Question 15

Define limsup (a_n)

Answer 15

limsup {n→∞} (a_n) = lim{k→∞} (sup{n≥k} (a_n))

Question 16

Define liminf (a_n)

Answer 16

liminf {n→∞} (a_n) = lim{k→∞} (inf{n≥k} (a_n))

Question 17

Equivalent Formulation

Answer 17

If two subsequences of a real sequence (a_n) converge to different limits, then (a_n) is not convergent.

lim (a_{n_k}) =/= lim (a_{n_k’})
k→∞ k’→∞