91Ó°ÊÓ

In mathematical induction, the first step is to establish the validity of the statement for the initial value of the variable, known as the 'base case'. This usually involves verifying the statement for the smallest possible value of the variable in question.
For example, to prove that the inequality \( n < n + 1 \) holds true, we start with the base case where \( n = 1 \).
Substituting \( n = 1 \) into the inequality, we get \( 1 < 1 + 1 \), which simplifies to \( 1 < 2 \). Since this is a true statement, the base case is satisfied.
This initial verification is crucial, as it serves as the foundation for the subsequent steps in the mathematical induction process.

The second step in mathematical induction is to make an assumption called the 'inductive hypothesis'. Here, we assume that the statement holds true for some arbitrary positive integer \( k \).
In the context of our example, we assume that \( k < k + 1 \) is true for a chosen value of \( k \).
This assumption is critical as it sets up the next step where we seek to prove that if the statement holds for \( k \), it must also hold for \( k + 1 \).
By assuming the inductive hypothesis, we are essentially preparing the groundwork to extend the validity of our statement beyond the base case.

The final step in mathematical induction is the 'inductive step'. This involves proving that the assumed statement (inductive hypothesis) also holds for the next integer, \( k + 1 \).
Continuing with our example, we start with the inductive hypothesis: \( k < k + 1 \). From this, we need to prove that \( k + 1 < k + 2 \).
We observe that adding 1 to both sides of the initial assumption \( k < k + 1 \) maintains the inequality, yielding \( k + 1 < k + 2 \).
Since this inequality holds true, the inductive step is satisfied.
With both the base case and the inductive step verified, we have successfully proven the statement \( n < n + 1 \) for all positive integers \( n \) using mathematical induction.