Proof by Mathematical Induction.

[Airaya]

Prove the following by induction on n:
∀ m ∈ N, ∀ n ∈ N( n ≥ m -> m|n!)
Hint: your "base" case will be at n = m.
I know what I want to show for the inductive step, I'm just having a difficult time writing it in proof form. So if we assume ∀ m ∈ N, ∀ n ∈ N( n ≥ m -> m|n!) is true then then n+1 case is true.
so n ≥ m ->m|(n+1)! I'm not sure if that when you show the n+1 case do you also show the right side of the implication as n+1?
So instead it would be n+1 ≥ m ->m|(n+1)! ?




I'm also unsure of how to do this one.
Prove the following by induction on n:
∀ m ∈ N, ∀ n ∈ N( m ≤ n -> m|n!)
Hint: (n + 1)! = (n + 1) (n!).
Hint: Carefully stating P(n) will make this much easier.
Note: I am aware that this sentence is equivalent to the previous problem., but the proofs are fairly
different, though.

 

[stapel]

Prove the following by induction on n:
∀ m ∈ N, ∀ n ∈ N( n ≥ m -> m|n!)
Hint: your "base" case will be at n = m.
I know what I want to show for the inductive step, I'm just having a difficult time writing it in proof form. So if we assume ∀ m ∈ N, ∀ n ∈ N( n ≥ m -> m|n!) is true then then n+1 case is true.
so n ≥ m ->m|(n+1)! I'm not sure if that when you show the n+1 case do you also show the right side of the implication as n+1?
So instead it would be n+1 ≥ m ->m|(n+1)! ?

I'm not sure what you're saying here...?

Taking as the "base" case "n = m", you have n! = 1*2*3*...*(n-1)(n), and m = n, so n! = 1*2*3*...*(m-1)(m) = 1*2*3*...*(m-1)*m. Letting q = (m-1)!, then we have n! = mq, so m divides n!.

I'm guessing the "next" case would be for n = m + k for some natural k > 0. You would assume that m divides n! Then you would consider n = (m + k) + 1.

What are you doing?

I'm also unsure of how to do this one.
Prove the following by induction on n:
∀ m ∈ N, ∀ n ∈ N( m ≤ n -> m|n!)
Hint: (n + 1)! = (n + 1) (n!).
Hint: Carefully stating P(n) will make this much easier.
Note: I am aware that this sentence is equivalent to the previous problem., but the proofs are fairly different, though.

Was this given as a separate exercise? What is "P(n)"? On what basis have you determined that the proof would be "fairly different"? What have you gotten as your proof for the other exercise?

Please be complete. Thank you!