III - Sequences and limits

1.1 Sequence

A sequence of real numbers is a function f : IN → IR

We use the notation a_n= f(n) for the elements of the sequence and (a_n) for the whole sequence

(a_n) = (a₁,a₂,a₃,…,a_n,…)

1.3 Convergence

A sequence (a_n) converges to a limit a ∈ IR if

for all ε > 0 there exists N ∈ IN such that if n > N then |a_n - a| < ε

so as a_n → a as n → ∞

every convergent sequence is bounded

1.8 Divergence

• a sequence is divergent if its not convergent

• a sequence is divergent to ∞ if for all M > 0 there exists N ∈ IN such that a_n > M for all n > N. We write a_n → ∞ and n → ∞

• a sequence is divergent to -∞ if for all M > 0 there exists N ∈ IN such that a_n < -M for all n > N. We write a_n → -∞ and n → ∞

2.1 Null sequence

(a_n) is a null sequence iff (|a_n|) is a null sequence

(convergent to zero but never hit zero e.g. (a_n) = 1/n)

2.4 The Shift Rule

let M ∈ IN

the sequence (a_n) is convergent iff the sequence (a_n+M) is convergent

lim a_n = lim a_n+M

2.6 Rules of limits

• c ∈ IR, if a_n → a then ca_n → ca

• if a_n → a and b_n → b then a_n + b_n → a + b

• if a_n → a and b_n → b then a_nb_n → ab

• if a_n → a and b_n → b then a_n/b_n → a/b

2.9 The Sandwich Rule

let a ∈ IR and let (a_n),(b_n),(c_n) be three sequences with a_n ≤ b_n ≤ c_n for all n ∈ IN

if a_n → a and c_n → a then b_n → a

3.1 Increasing / decreasing sequences

• A sequence (a_n) of real numbers is increasing if a_n ≤ a_n+1 for all n

• A sequence (a_n) of real numbers is decreasing if a_n+1 ≤ a_n for all n

• A sequence is monotone if it is neither increasing or decreasing

• if a sequence (a_n) is increasing and bounded above then it is convergent : lim a_n = sup a_n

• if a sequence (a_n) is decreasing and bounded below then it is convergent : lim a_n = inf a_n

4.1 Subsequence

let n_k be a strictly increasing sequence of natural numbers

then a_ⁿk is called a subsequence of a_n

lim a_n = lim a_ⁿk

_{every sequence has a monotone subsequence}

4.6 Bolzano-Weierstrass theorem

every bounded sequence has a convergent subsequence

4.8 Accumulation point

let (a_n) be a bounded sequence

we call a point a ∈ IR an accumulation point of the sequence if there exists a subsequence a_ⁿk that converges to a

the smallest accumulation point is the limit inferior, denoted by lim inf a_n

the largest accumulation point is the limit superior, denoted by lim sup a_n

if (an) is a bounded sequence then lim inf a_n = lim inf a_k , k>n and lim sup a_n = lim sup a_k , k>n

4.11 Convergent in terms of lim sup / lim inf

let (an) be a bounded sequence

then (an) is convergent if and only if lim inf a_n = lim sup a_n which equals lim a_n

5.1 Cauchy sequence

A sequence (a_n) of real numbers is a Cauchy sequence if

for all ε > 0 there exists N ∈ IN such that if n,m > N then |a_n - a_m| < ε

every convergent sequence is a Cauchy sequence

every Cauchy sequence is bounded

5.5 Cauchy’s Convergence Criterion

every Cauchy sequence of real numbers is convergent

6.1 Bernoulli’s inequality

for all n ∈ IN₀ and for all x > -1, (1 + x)ⁿ ≥ 1 + nx