I tried proving the proposition by induction. Here is my proof that I received a
ID: 3075706 • Letter: I
Question
I tried proving the proposition by induction. Here is my proof that I received a 0/3 points for:Definition: m>n if m-n>0.
Induction on n: P(n): m>n.
(i) If n=1, m ? n+1 must exist and m ? N for P(1) to be true.
n=1 ? (1)+1=2. m ? 2, 2 ? N, thus P(1) holds.
(ii) Assume n ? N and P(n) holds
This implies (n+1) + 1 = n+2. m ? n +2, n+2 ? N, thus P(n+1) holds.
? P(n) ? P(n+1).
All I got for feedback from my professor is an underlining of "P(n): m>n." and question marks in line 2.
Explanation / Answer
My feedback: At (ii), instead write, "Assume k ? N and P(k) holds" Because "P(n) holds" is what you're trying to prove and you can't assume what you're trying to prove. Then show that P(k+1) holds. Your part (i) is correct. Here's how I would do (ii): Assume k ? N and P(k) holds. Then for some m, m > k. So m+1 > k+1. Since m+1?N, P(k+1) holds. ? P(k) ? P(k+1).