I'm trying to prove by induction the following statement without success:
$$\forall n \ge 2, \;\forall d \ge 2 : d \mid n(n+1)(n+2)...(n+d-1) $$
For the base case: $n = 2$, $d = 2$
$2\mid 2(2+1)$ which is true.
Now, the confusion begins! I assume I would need to use the second induction principle to proof this because $P(n)$ and $P(n+1)$ are not related at all. It is also the first time I am dealing with more than one variable so it makes it harder for me.
I tried the following:
- Trying to prove by simple induction. I did not go very far.
- Trying to split my induction step in 2 parts: If $d\mid n$, I'm done.
If $d$ does not divide $n$, then I would need to do a second proof and this is where I'm blocked.
Anyone could tell me what's wrong in the approach I take to solve this problem?
Any help would be appreciated!