Sequences of Mathematical Statements

In mathematics, a sequence is an ordered list of objects, or elements. The length of a sequence is the number of ordered elements, and it may be infinite. Unlike a set, order matters in sequences, and exactly the same elements can appear multiple times at different positions in the sequence. A sequence is a discrete function. For example, $\left ( M,A,R,Y \right )$ is a sequence of letters that differs from $\left ( A,R,M,Y \right )$. Although the composition is the same, the ordering differs. Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers $(2,4,6, \cdots )$. 

In mathematics, a "sequence of statements" refers to the progression of logical implications of one statement. In this case, a "statement" usually refers to an equation that contains an equal sign. Sequences of statements are necessary for mathematical induction. Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true for all natural numbers. It is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if any one statement in the infinite sequence of statements is true, then so is the next one.

For example, in the context of mathematical induction, a sequence of statements usually involves an algebraic statement into which you can substitute any natural number $(0, 1, 2, 3, ...)$ and the statement should hold true. So a sequence is formed by substituting integers $k$, $k + 1 $, $k + 2$ and so on into the mathematical statement. This concept will be expanded on in the following concept, which introduces proof by mathematical induction.