Proving the Inductive Step of an Inductive Proof

The inductive step is the engine that drives proof by induction. It's what connects one case to the next, creating the "domino effect" that allows us to prove a statement for infinitely many numbers.

Here's how it works:

The Assumption (Inductive Hypothesis): We assume that our statement is true for some arbitrary number, which we usually call $k$ . This is like assuming one of the dominoes has fallen.
The Proof: Using this assumption, we then prove that the statement must also be true for the next number, $k+1$ . This is like showing that if one domino falls, it will knock over the next one.

If we can do this, we've created a chain reaction. If the statement is true for $k$ , it must be true for $k+1$ . This, combined with the base case, proves the statement for all numbers greater than or equal to the base case. In this example the 'next' number after $k$ is $k+1$ , but this relationship depends on how we set up our proof as explained in this article about inductive strategies

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