The Proof by Induction Hypothesis is a mathematical proof technique used to prove that a statement holds true for all natural numbers. It involves three steps: the base case, the inductive hypothesis, and the inductive step.
The base case is the first step in the proof by induction. It involves showing that the statement holds true for the first natural number, typically 0 or 1. This serves as the starting point for the proof.
The inductive hypothesis is the second step in the proof by induction. It assumes that the statement holds true for some natural number k, and then uses this assumption to prove that it also holds true for the next natural number, k+1. This is the key step in the proof by induction.
The inductive step is the final step in the proof by induction. It involves using the inductive hypothesis to show that if the statement holds true for some natural number k, then it also holds true for the next natural number, k+1. This completes the proof by showing that the statement holds true for all natural numbers.
Some common mistakes to avoid when using Proof by Induction Hypothesis include: not clearly stating the base case, assuming that the statement holds true for all natural numbers without proving it, and using circular reasoning. It is important to carefully follow the steps of the proof by induction and clearly explain each step to avoid these mistakes.