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I was reading the post here and I understand how the method works. However, I'm not sure why the proof qualifies as a proof by induction. Based on my understanding, proof by induction works like this:

(1) Show that it is true when $n = 1$;

(2) Show that it is true when $n = k$;

(3) Show that it is true when $n = k+1$.

However, when the answer derives this equation: \begin{align}V(T_n(1))&=\int_{x_n\le 1}\left(\int_{x_1+\cdots+x_{n-1}\le1-x_n}dx_1\cdots dx_{n-1}\right)dx_n\\ &=V(T_{n-1}(1))\int_{x_n\le1}\color{red}{(1-x_n)^{n-1}}dx_n=\frac1 nV(T_{n-1}(1))\end{align}

I'm not sure how it was inducted. I wrote out the cases when $n = 1,2,3$, but I could not prove a general case where $n = k$. Can someone please give me a hint on (1) how to prove the general case when $n = k$, and (2) do we actually need to prove the case where $n = k$?

user1691278
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  • Your idea of induction doesn't sound right. In the most common form, you assume it for $n=1$ or some base case, then assume it is true for $n=k$ and use it to show that it is true for $n=k+1$. – OnceUponACrinoid Apr 10 '18 at 06:49
  • @OnceUponACrinoid This is where I learned about proof by induction: http://comet.lehman.cuny.edu/sormani/teaching/induction.html. Regardless of that, how does the post prove the equation by induction? – user1691278 Apr 10 '18 at 06:51

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