Discrete Mathematics Proofs for ∃ and ∀ Premises or Givens:

$∃x(A(x) → B(x))$
$∀x (B(x) → K(x))$

To Prove:

$∃x(A(x) → K(x))$

My Solution:

$A(z) → B(z)$  From premise and  Existential instantiation $x$ for $z$

$B(z) → K(z)$  From premise and  Universal instantiation $x$ for $z$

$A(z) → K(z)$  Transitivity of 1,2

$∃x(A(x) → K(x))$  From Existential generalization (Substitute $z$ for $x$)


OR
I was thinking about assuming $A(z)$ and then using Modus Ponens to get $B(z)$ and then further $K(z)$, then using the deduction theorem on $A(z)$ and $K(z)$, and then using Existential Generalization on that statement by substituting $x$ for $z$.

Can someone suggest which way would be more effective?
Is there any other effective way to solve it?
Your solution is a good one, and that's how I'd prove it also. I don't believe a more straightforward or simpler version is possible.