Let $$\begin{eqnarray*} A(0,y) &=& y+1 \\ A(x+1,0) &=& A(x,1) \\ A(x+1,y+1) &=& A(x,A(x+1,y)) \end{eqnarray*}$$
I want to prove by induction over x that $$A(x,y) < A(x,y+1) \; \forall x,y \in \mathbb{N} $$
Basis: $x = 0$
$$ A(0,y) = y+1 < y+2 = A(0,y+1)$$
So far so easy, but I'm completely stuck with the induction step. Could you please help me to go on?
Thanks in advance!