Week 4

Question 1

real sequence

Answer 1

A real sequence is a function N → R

Question 2

Let (an)^∞_n=1 be a real sequence and let L ∈ R. We say that an converges to L as n tends to infinity if and only if

Answer 2

for each ε > 0, there exists n0 ∈ N, such that for n ∈ N with n ≥ n0, we have |an − L| < ε.

Question 3

When an converges to L as n tends to infinity, we write

Answer 3

“an → L as n → ∞“ or limn→∞ an = L

Question 4

We say that the sequence (an)^∞_n=1
diverges if

Answer 4

it does not converge to any limit.

Question 5

The definition can be written concisely as a quantified statement:
the sequence (an)^∞_n=1 converges to L ∈ R if and only if

Answer 5

∀ε > 0, ∃n0 ∈ N s.t. ∀n ∈ N,(n ≥ n0 =⇒ |an − L| < ε).

An alternative form is

∀ε > 0, ∃n0 ∈ N s.t. ∀(n ∈ N with n ≥ n0), |an − L| < ε.

Question 6

“|an − L| < ε” says that

Answer 6

an is within distance ε
of L”

Question 7

prove that certain sequences converge directly from the definition

Answer 7

you must verify the condition in definition 3.2 directly, without using subsequent results.

Question 8

constant sequences

Answer 8

Sequences of the form x_n = K for all n ∈ N