Questions about Probability: Theory and Examples problem 5.3.8: why do we have $\lambda<x(1-p)$

Question

I was doing Probability: Theory and Examples problem 5.3.8 To use Theorem 5.3.8, we need to prove $\mathbb{E}_x[\phi(X_1)]<\phi(x)$, which is just $\mathbb{E}_x[X_1]<x$. After some simple calculations, we have $\mathbb{E}_x[X_1]=px+\lambda$, so we must have $\lambda<x(1-p)$ for any $p<1$.

My question is, why do we have $\lambda<x(1-p)$? I think I must have something not understand here, but why is this true?

Thank you!

Answer 1

If I read Theorem 3.5.8 correctly then $\mathbb E_x[\varphi(X)]\le x$ can be violated for finitely many $x$. Here clearly the inequality holds true for $x\ge\frac{\lambda}{1-p}$, which is fine as the set $\{0,1,\cdots,\lfloor\frac{\lambda}{1-p}\rfloor\}$ is finite.