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Every mathematical induction consists of three parts or acts.

The first part is called “The Base”. The mathemagician shows you something ordinary: a claim, a theorem or a lemma. He asks you to see if it is indeed real, unaltered, true for k = 1.

The second act is called “The Hypothesis”. The mathemagician takes the ordinary claim and assumes it to hold for k = n.

Now you’re looking for the trick … but you won’t find it, because of course you’re not really looking. You don’t really want to know. You want to be fooled. But you wouldn’t clap yet. Because assuming something for k = n isn’t enough. You have to prove it for k = n + 1.

That’s why every induction proof has a third act, the hardest part, the part we call “The Inductive Step”.

—Sanchit Agrawal (via michaelblume)

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