Ordered variables
When referring to the independent variable x, we have only been concerned with the set of the values that x can assume. For example, this can be the set of values satisfying 0 < x < 1. We shall now consider the variable x taking an infinity of values in sequence, i.e. we are now interested, not only in the set of values x, but also in the order in which it takes these values. More precisely, we assume the possibility of distinguishing, for every value of x, a value that precedes it and a value that follows it, it being also assumed that no value of the variable is the last, i.e. whatever value we take, there exists an infinity of successive values.
A variable of this type is sometimes called ordered. If x', x" are two values of the ordered variable x, a preceding and a succeeding value can be distinguished, whilst if x' precedes x", and x" precedes x'", then x' precedes x"' . We shall assume, for example, that the set of values of x is defined by 0 < x < 1, and that of two distinct values x' and x" the succeeding value is the greater. We thus obtain an ordered variable, continuously increasing through all real values from zero to unity, without reaching unity. The sequence of values of the variable, for phenomena occurring in time, is established by the temporal sequence, and we shall sometimes make use of this time-scheme below, using terms such as "previous" and "later" in place of "preceding" and "succeeding" values.
An important particular case of an ordered variable is that when the sequence of values of the variable can be enumerated, by arranging them in a series of the form:
so that, given two values 
the value succeeds that has the greater subscript. In the case mentioned above, when the variable increases from zero to unity, we can clearly not enumerate its successive values. It may also be noted that it is possible to encounter identical values amongst those of an ordered variable. For example, we might have 
in the enumerated variable. Abstracting, as we always do, from the concrete nature of the magnitude (length, weight etc.), we must understand by the term "ordered variable", or as we shall say for brevity, "variable", simply the total sequence of its numerical values. We normally introduce one letter, say x, and suppose that it assumes successively the above-mentioned numerical values.
the value succeeds that has the greater subscript. In the case mentioned above, when the variable increases from zero to unity, we can clearly not enumerate its successive values. It may also be noted that it is possible to encounter identical values amongst those of an ordered variable. For example, we might have 
in the enumerated variable. Abstracting, as we always do, from the concrete nature of the magnitude (length, weight etc.), we must understand by the term "ordered variable", or as we shall say for brevity, "variable", simply the total sequence of its numerical values. We normally introduce one letter, say x, and suppose that it assumes successively the above-mentioned numerical values.
For every value of the variable x, a corresponding point K is defined on the axis X. Thus, as x varies in sequence, the point K moves along X. The present paragraph is devoted to the basic theory of limits, which is fundamental to all modern mathematical analysis. This theory considers some extremely simple, and at the same time, extremely important, cases of variation of magnitudes.
the source:
A COURSE OF Higher Mathematics VOLUME I. SMIRNOV.
By: Fady tarek
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