Question

given property \\(p = p(n)\\), prove that it holds for all integers \\(n \ge 0\\).

Explanation:

Identify the proof method

To prove a property \(P(n)\) holds for all integers \(n \ge 0\), we use Mathematical Induction.

Establish the base case

Verify that the property holds for the smallest integer in the set.
Show that \(P(0)\) is true.

State the inductive hypothesis

Assume the property holds for an arbitrary integer \(k \ge 0\).
Assume \(P(k)\) is true.

Perform the inductive step

Prove that the property holds for the next integer \(k + 1\).
Show that \(P(k) \implies P(k+1)\).

Conclude the proof

By mathematical induction, \(P(n)\) holds for all integers \(n \ge 0\).

Answer:

To prove that a property \(P(n)\) holds for all integers \(n \ge 0\), use the Principle of Mathematical Induction by completing the following two steps:

1. Base Case: Prove that the property holds for the initial value, \(n = 0\). That is, show that \(P(0)\) is true.
2. Inductive Step: Assume that the property holds for an arbitrary integer \(k \ge 0\) (this is the inductive hypothesis, assuming \(P(k)\) is true). Then, prove that the property must also hold for the next integer, \(n = k + 1\). That is, show that \(P(k) \implies P(k+1)\).

Once both steps are successfully completed, the property \(P(n)\) is proved to be true for all integers \(n \ge 0\).