A sequence can be defined as a mapping between a non-empty set S and the subset of natural numbers, always denoted as (S_n). This set's co-domain represents the image of the nth term of the set of natural numbers N.

Range of Sequence:

The set comprising all distinct elements of a sequence is known as its range. It is denoted by (S_n) = ((-1)^n : n ∈ N) = (-1, 1, -1, ...) with its range being (-1,1).

Types of Sequence:

Constant Sequence:

A sequence where the range set only has one element is referred to as a constant sequence. For example, (S_n) = ((1)^n : n ∈ N) = (1, 1, 1, ...) with its range being (1,1).

Increasing Sequence:

A sequence is an increasing sequence when (S_n) comprises values with s_(n+1) ≥ s_n for all n ∈ N. An example of this sequence could be (S_n) = (1, 2, 3, ...).

Decreasing Sequence:

A sequence is known as a decreasing sequence when (S_n) comprises values with s_(n+1) ≤ s_n for all n ∈ N. An example of this sequence could be (S_n) = (-1, -2, -3, ...).

Subsequence:

For any sequence (s_n), a sequence of positive integers (n_k), that is strictly increasing, denotes that (s_nk) is a subsequence of (s_n). For instance, (S_n) = (s_1, s_2, s_3, s_4, s_5, ...) whereby (S_nk) = (s_3, s_4, s_5, ...) is considered as a subsequence of (S_n).

Convergent Sequence:

When a sequence (S_n) is said to converge to 'l' belonging to R, if for any given arbitrary Ɛ > 0, there exists m ∈ N such that |S_n - l| < Ɛ for all n ≥ m. This is also denoted as lim S_n = l.

Important Result on Convergent Sequence:

1. The limit on a convergent sequence is unique.

2. Any subsequence of (S_n) converges to the same limit 'l' as the original sequence if (S_n) converges to 'l'.

3. Every convergent sequence is bounded.

Divergent Sequence:

1. A sequence (S_n) is said to diverge to +∞ if there exists a non-ever-large K > 0 such that S_n > K for all n ≥ m.

2. Similarly, if there exists a non-ever-small K < 0 such that S_n < K for all n ≥ m, then the sequence (S_n) is said to diverge to -∞.

Oscillatory Sequence:

A sequence is categorized as an oscillatory sequence when it neither converges nor diverges.

Monotonic Sequence:

A sequence is known as a monotonic sequence when it is either monotonically increasing or monotonically decreasing.

Monotonically Increasing:

A sequence is referred to as monotonically increasing when s_(n+1) ≥ s_n for all n.

Monotonically Decreasing:

A sequence is monotonically decreasing when s_(n+1) ≤ s_n for all n.

Important Result on Monotonic Sequence:

Every bounded and monotonically increasing sequence is convergent.

Cauchy Sequence:

In the realm of real analysis, sequences play a fundamental role. A sequence is a type of function with a domain set comprising of natural numbers. The key takeaway is that a sequence is a tool to understand various aspects of mathematics.

A Cauchy sequence, in particular, has important properties. Firstly, it is bounded. Secondly, a real sequence is said to be convergent only if it is a Cauchy sequence. This is a critical finding.

Another type of sequence is the nested sequence of sets. This sequence (A_(n)) involves sets ordered with the subset symbol ‘>’ such that A₁>A₂>A₃>A₄……….>A_(n)>A_(n+1) ....

There are different concepts related to sequences, such as convergences, divergence, and limit point. These topics serve as tools in our daily lives, helping us to understand different phases of real mathematics.