Blackwells Theorem in renewal theory and implications

Question

I am reading the renewal theory chapter from Sheldon Ross's book. There a non-negative random variable $X$ is said to be lattice if there exists $d \geq 0$ such that $\sum_{n=0}^\infty P \{X=nd\}=1 $. The largest $d$ having this property is called the period of $X$. For a renewal process where the interarrival distribution is such a $X$ with mean $\mu$, Blackwells Theorem states that $$\mathbb E[\text{number of renewals at $nd$}]\to \frac{d}{\mu} \quad \text{as} \ n \to \infty$$ The book then says that this implies $$\lim_{n \to \infty}P(\{\text{renewal at nd}\})=\frac{d}{\mu}$$ I don't see how this follows from the theorem. Any help will be appreciated.