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I figured that for any induction that I write down I can just use a t = n+1 and the entire question becomes as easy as doing 1+1. and obviously it cant be legal to do so, but I wonder why, can someone explain? example : Prove by induction that for all n≥3: n^(n+1)>(n+1)^n lets just take a t = n+1, then (n+1)^(n+2) > (n+2)^(n+1), can be written down as t^(t+1)>(t+1)^t and boom it solved itself?

Ravid Ofek
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    But $t>n$ you are assuming the truth for $k\le n$. – David P Nov 01 '22 at 11:24
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    For some basic information about writing mathematics at this site see, e.g., here, here, here and here. – Another User Nov 01 '22 at 11:33
  • Yes, if $t=n+1$, then the inequality $(n+1)^{n+2}>(n+2)^{n+1}$ becomes $t^{t+1}>(t+1)^t$. And… why “boom”? That is, why would this be obviously true? – José Carlos Santos Nov 01 '22 at 11:35
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    The induction hypothesis assumes the proposition is true for a fixed (but unknown) $n$. It's not assuming that it works for all $n$ (that would be assuming what you wish to prove). The idea is that you start proving the base case (often $n = 1$). The induction step with $n = 1$ subbed in will give you a proof that, because the $n = 1$ case holds, so does the $n = 2$ case. Then, using the induction step again, you then know that the $n = 3$ case holds. Then $n = 4$, $n = 5$, and so on. You only assume one case at a time, so you can get to the next case. – Theo Bendit Nov 01 '22 at 12:14

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