Inequalities (Natural Numbers)

Question

Suppose that $1 < x$ and $z < x^{z}$ is true where $x, z \in \mathbb{N}$. Prove that $z + 1 < x^{z+1}$. I have tried to use every inequality but have not been able to find the proof. This proof is part of a proof by mathematical induction.

Answer 1

I believe that the answer above is correct with a small correction $$ x^{z+1} =x^z \times x \ge x^z \times 2 >(z+1) \times2 \ge z+1 $$

If $$ z=0 $$ then $$ x^{z+1}>z+1 \Longleftrightarrow x^{0+1}> 0+1 \Longleftrightarrow x>1 $$ valid. If z>0 then we already know that $$ z + 1 < x^{z+1} $$