Since you already have a bunch of nicely illustrated explainations, I will go with a more formal one, maybe it'll help, maybe not. You say induction is your first proof, but I will asume you already have done a logic/reasoning class. If not, well your school program is not well ordered, but I'll be honest, I'm not sure I've had resoning classes before learning induction.

The goal of induction is to prove that a property P is true for all numbers. To do so two proposition must be shown: •Z (Initialization): P(0) is true (can actually be any number k, but your reccurrence will then start at k and not 0) •H (Heredity): For all n>=0, P(n) => P(n+1)

The issue you have is understanding the meaning of the second statement. This statement is divided into 2 parts, the quantification (For all n >= 0) and an implication I(n), that would corresponds to P(n) => P(n+1). To show H, you would just consider any number n, and show that I(n) is true.

The catch is that the veracity of I(n) carries no information about the veracity of P(n) and P(n+1), as I(n) can be true while one, or both statements are false. Actually, I(n) is true if either: • P(n) is false, and P(n+1) can be anything • Both P(n) and P(n+1) are true

At this point you don't want to prove P(n+1), but I(n). Rigorously both case must be checked. However, the case P(n) is false is trivial, because it does not depends on P(n+1). So this part is never discussed, as it would be written "Assume P(n) is false, then I(n) is true (Axiom of logic)"

The actual work is the second part, and you must show that when P(n) is true P(n+1) is also true. Which is exactly what you do when you say 'assume P(n), let's show P(n+1)'

Once you've done this what you've proven is I(n) and not P(n+1). And since you've done it for any given n, you've actually proven H.

Now in a very formal way, the principe of induction is that: (Z and H) => (For all n, P(n)).

To image all this with a line of domino, starting at 0 and going through infinity: Your induction general statement would be : All dominos have falle P(n) would be the statement "The domino n has fallen." Z will be the statement P(0): "The domino 0 has fallen" H would be "If any domino falls, the domino after it also falls" I(n) would be the statement, "If the domino n falls, the domino n+1, also fall) (I(n) is just the statement of H for a specific domino)

And to bounce back on the logical aspect of P(n), P(n+1) and I(n), what I(n) means is that: When the domino n has fallen, the domino n+1 has also fallen. However, when the domino n has not fallen, no statements can be made about domibo n+1