Problem with understanding the proof of Sauer Lemma

[yongxien]

I will replicate the proof here which is from the book "Learning from Data"

Sauer Lemma:
$B(N,K) \leq \sum_{i=0}^{k-1}{n\choose i}$

Proof:
The statement is true whenever k = 1 or N = 1 by inspection. The proof is by induction on N. Assume the statement is true for all $N \leq N_o$ and for all k. We need to prove that the statement for $N = N_0 + 1$ and fpr all k. Since the statement is already true when k = 1(for all values of N) by the initial condition, we only need to worry about $k \geq 2$. By (proven in the book), $B(N_0 + 1, k) \leq B(N_0, k) + B(N_0, k-1)$ and applying induction hypothesis on each therm on the RHS, we get the result.

**My Concern** From what I see this proof only shows that if $B(N, K)$ implies $B(N+1, K)$. I can't see how it shows $B(N, K)$ implies $B(N, K+1)$. This problem arises because the $k$ in $B(N_0 + 1, K)$ and $B(N_0, K)$ are the same, so i think i need to prove the other induction too. Why the author is able to prove it this way?

[yongxien]

OK i think i will just post it below. I can't find an edit button. I mean for 2 variable induction, shouldn't we prove B(N,k) implies B(N+1,k) and B(N, K+1)?

[htlin]

You can imaging that the induction hypothesis to be satisfying the inequality for "all k", and then, satisfies the inequality for "all k" too.

Hope this helps.

[yongxien]

That is my concern. Why "all k" when we have not proved it.