12. Prove that if n >m then the number of m-cycles in Sis given by nn-1)(n-2)......
Problem 7: Prove that for all integers n > 2, n+1 n 10-11 - n n +
Prove that is an integer for all n > 0.
Use the Principle of Mathematical Induction to prove that (2i+3) = n(n + 4) for all n > 1.
2. Prove that if n > 1, then 1(1!) + 2(2!) + ... + n(n!) = (n + 1)! - 1.
Problem 30. Prove that N, Z, Q and R are infinite sets. (HINT: Prove by induction on n that is f: NN then (3k N(Vj Nn)k> f(j). Then conclude that f cannot possibly be onto N. A similar strategy works for Z, gq and R as well.)
Prove by mathematical induction (discrete mathematics) n? - 2*n-1 > 0 n> 3
Problem 5.1.3. Prove by induction on n that (1+ n < n for every integer n > 3.
Please Prove. Prove 2 n > n2 by induction using a basis > 4: Basis: n 5 32> 25 Assume: Prove:
.n= n(n-1)(n+1) for all n > 2. 12. Use induction to prove (1 : 2) +(2-3)+(3-4) +...+(n-1).n [9 points) 3
For all n E N prove that 0 <e- > < 2 k!“ (n + 1)! k=0 Hint: Think about Taylor approximations of the function e".