Sequence of real numbers and related terminologies with examples

Sequence of real numbers:    

A function whose domain is the set of natural number and range is the set of the real number is defined as the sequence of real numbers. i.e. any function \(f:N \to R\) can be said to real sequence. In general language sequence can be understood as a collection of objects or events with some kind repetition or similarity. In Real Analysis, the concept of sequence is vaster, but the overall linguistic view is a little bit similar. In this article, the definition of sequence, bounds of sequence and the well-known Bolzano weierstrass property is discussed with simple examples.
Sequences are denoted by \(\left\{ {{S_n}} \right\}\) , for example \(f(x) = {x^2}\) defined from \(N\) to \(R\) can be considered  as a sequence. More conveniently, the sequence can be written as

\({S_n} = \{ {n^2}\}  = \{ {1^2},{2^2},{3^2},....\}  = \{ 1,4,9,....\} \) 

A sequence may have finite number of points, called finite sequence and may have or infinite number of points, called infinite sequence. The characteristic of a sequence can be determined by how bigger or how smaller its elements are. Therefore, the concept of boundedness is evolved in sequence which is elaborated below.

Bounds of a sequence:

A sequence \(\left\{ {{S_n}} \right\}\) is said to be bounded above if there exist a number \(k\)such that
\({S_n} \le k\)for all \(n \in N\)

For example: Let us consider the sequence \(\{ {S_n}\} \), where \({S_n} = \frac{1}{n}\)
Therefore, the terms of the sequence are \(\{ {S_n}\}  = \left\{ {1,\frac{1}{2},\frac{1}{3},..} \right\}\)
Here, \({S_1} = 1,{S_2} = \frac{1}{2},{S_3} = \frac{1}{3},..,{S_n} = \frac{1}{n},...\)
By simply looking at the terms we can see that  \(1,\frac{1}{2},\frac{1}{3},.. \le 1\)
i.e. \({S_1},{S_2},{S_3},..,{S_n},... \le 1\)
or we can say that \({S_n} \le 1\)for all \(n \in N\)
Therefore, the sequence \(\left\{ {{S_n} = \frac{1}{n}} \right\}\)is said to be bounded above.

A sequence \({S_n}\) is said to be bounded below if there exist a real number \(k\)such that
\(k \le {S_n}\)for all \(n \in N\)

For example: Let us consider the sequence \(\{ {S_n}\} \), where \({S_n} = {n^2}\)
Therefore, the terms of the sequence are \(\{ {S_n}\}  = \left\{ {1,4,9,..} \right\}\)
Here, \({S_1} = 1,{S_2} = 4,{S_3} = 9,..,{S_n} = {n^2},...\)
By simply looking at the terms we can see that  \(1 \le 1,4,9,..\)
i.e. \(1 \le {S_1},{S_2},{S_3},..,{S_n},...\)
or we can say that \(1 \le {S_n}\)for all \(n \in N\)
Therefore, the sequence \(\left\{ {{S_n} = {n^2}} \right\}\)is said to be bounded below.
More generally, a sequence is said to be bounded if the sequence is both bounded above and bounded below.

Limit point of a sequence:

A number \(\xi \)  is said to be a limit point of a sequence \(\left\{ {{S_n}} \right\}\) if every open interval consisting the point \(\xi \) contains an infinite number of elements of the sequence.
In a more general and simple way \(\xi \)  is said to be a limit point of a sequence \(\left\{ {{S_n}} \right\}\) if  \(\left( {\xi  - \varepsilon ,\xi  + \varepsilon } \right)\) contains an infinite number of points of the sequence, where \(\varepsilon \) is any positive number.

For example, let us consider the sequence \(\{ {S_n}\}  = \left\{ {1,\frac{1}{2},\frac{1}{3},..} \right\}\)
We consider \(\xi  = 0\) and \(\varepsilon \) be any positive number.
Then the open interval \(\left( {0 - \varepsilon ,0 + \varepsilon } \right) = \left( { - \varepsilon , + \varepsilon } \right)\)contains an infinite number of elements of the sequence \(\left\{ {{S_n}} \right\}\).
Therefore, \(\xi  = 0\) is a limit point of the sequence\(\left\{ {{S_n}} \right\}\).

Now, obviously one question may arise in the mind of the readers. “Does every sequence have limit points?” The answer of the question lies in the most fundamental theorem of the real sequence. The theorem is known as Bolzano weierstrass theorem. This theorem relates the existence of the limit point of a sequence with the boundedness of the sequence. The statement of the theorem is given below:

Bolzano weierstrass theorem:

Every bounded sequence has a limit point.
In a simple way, we can say that a sequence having its upper and lower bound must have a limit point.

For example, let us consider the sequence \(\{ {S_n}\} \), where \({S_n} = \frac{1}{n}\)
Here, \({S_1} = 1,{S_2} = \frac{1}{2},{S_3} = \frac{1}{3},..,{S_n} = \frac{1}{n},...\)
By simply looking at the terms we can see that  \(0 < 1,\frac{1}{2},\frac{1}{3},.. \le 1\)
i.e. the sequence is bounded above by 1 and bounded below by 0. Thus, the given sequence is bounded.
Therefore, by Bolzano weierstrass theorem one can say that the sequence has a limit point.
The limit point is 0, which is already shown in the earlier example.

In this article, the definition of real sequences and its related terminologies (limit point of a sequence, the bounds of a sequence and the well-known Bolzano weierstrass property or Bolzano weierstrass theorem) are discussed with examples. For each terms, one example is given for good understanding about the terms related to sequences.