Sequence

Finite Sequences- A function whose domain is the first n natural number, 
i.e., {i ∈ ℕ : i≤n}.

Infinite Sequences- A function whose domain is the set of natural numbers.

Countable Set- Set is countable if it the range of some sequences.

Finite Countable Set- Set is finite countable if it is a range of come finite sequences.

Monotone Function- g :  ℕ → ℕ if  i > j  ⇒ g(i) > g(j) or g(i) < g(j).

Sequences of Real number:

Sequence- <x_n> of real number is a function whose domain is the set of natural numbers.
Limit- A real number l is a limit of a sequence <x_n>, lim x_n = l
 if ∈ > 0, ∃ N : ∀ n ≥ N, | x_n - l| < ∈.
Cauchy sequence- Sequence <x_n> is cauchy sequence if ∈ > 0, ∃ N such that
∀ n ≥ N and ∀ m ≥ N we have | x_n-x_m| < ∈.
Convergent sequence- A sequence is called convergent sequence if it has a limit.

Divergent sequence- A sequence which is not a convergent sequence is called a divergent sequence.

Cluster point- A real number l is a cluster point of the sequence  <x_n> if for given 
 ∈ > 0 and given N, ∃ n ≥ N : | x_n - l| < ∈.

Some Important Results-

1. A sequence of real number is convergent iff it is Cauchy sequence.
2. If the limit of the sequence exist it is unique.
3. Every convergent sequence is bounded.
4. If a sequence <x_n> converges to l, then its subsequence also converges to l.

Sequence-  It is a function whose domain is the set of positive integers.

                       i.e. a_n = a(n), n = 1, 2, 3, ...

Let p_n be the nth prime number,

                             p_n = p(n), n = 2, 3, 5, ...

Convergence of a sequence- A sequence {a_n} converges to real number A iff for each 

∈ > 0, there is a positive integer N, such that ∀ n ≥ N, we have | x_n - A| < ∈.

Neighbourhood- A set N_x of real numbers is a neighbourhood of a real number x iff N_x contains an interval of positive length centered at x, 

i.e., iff there is ∈ > 0 : (x -∈, x +∈) ⊂ N_x.

Accumulation point- For a set S of real numbers, a real number A is an accumulation point of S iff every neighbourhood of A contains infinitely many points of S.

Limit of a sequence- If a sequence is convergent, the unique number to which it converges is the limit of a sequence.
Subsequence- Let {a_n} be a sequence and (n_k) be any sequence of positive integers such that
n₁ < n₂ < n₃ < . . . The sequence {a_nk} is called a subsequence of {a_n} for all n = 1, 2, . . . , ∞
Increasing sequence- Sequence {a_n} : n = 1, 2, . . . , ∞ is increasing, iff a_n ≤ a_{n + 1} for all n.
Decreasing sequence- Sequence {b_n} : n = 1, 2, . . . , ∞ is decreasing, iff b_n ≥ b_{n + 1} for all n.
Monotone sequence- Sequence that is either increasing or decreasing.
Bounded above sequence- Sequence {a_n} : n = 1, 2, . . . , ∞ is bounded above, iff there exists a real number
N : a_n ≤ N for all n.
Bounded below sequence- Sequence {a_n} : n = 1, 2, . . . , ∞ is bounded below, iff there exists a real number
M : a_n ≥ M for all n.
Bounded sequence- Sequence {a_n} : n = 1, 2, . . . , ∞ is bounded, if it is bounded both from above and below
⇔ there exists a real number S : |a_n| ≤ S for all n.

Some important theorems-

1. Every convergent sequence  is a Cauchy sequence.
2. Every Cauchy sequence is bounded.
3. Every Cauchy sequence is convergent.
4. A sequence is cauchy iff it is convergent.
5. A sequence converges iff each of its subsequences converges.
6. A monotone sequence is convergent iff it is bounded.
7. If {a_n} : n = 1, 2, . . . , ∞ converges to real number A and B, then A = B.
8. If {a_n} : n = 1, 2, . . . , ∞ converges to A, then {a_n} : n = 1, 2, . . . , ∞ is bounded.