Let $(\mu_n)_{n\in\mathbb{N}}$ be a sequence of probability measures on $\mathbb{N}$, such that the Laplace transform $\phi_n(\lambda)=\int e^{-\lambda x}\mu_n(dx)$ converges pointwise to a limit $\phi(\lambda)=\int e^{-\lambda x}\mu(dx)$ for some probability measure $\mu$ and $\lambda$ in some non-empty interval $(a,b)$. Prove that $\mu_n$ converges weakly to $\mu$.
Hint: Fix $\lambda_0\in(a,b)$ and rewrite $\phi_n$ as Laplace transform of the new probability measure $\eta_n(dx)=e^{-\lambda_0x}\frac{\mu_n(dx)}{\phi_n(\lambda_0)}$, with $\eta$ defined similarly. Show that $\eta_n$ is tight and converges weakly to $\eta$ and then deduce that $\mu_n$ converges weakly to $\mu$.
This is the continuity theorem with respect to Laplace transform. I have found it in "An introduction to Probability" by Feller. But it doesn't provide proofs clearly. Does anyone see the proof of the the continuity theorem with respect to Laplace transform? Can you recommend me the source about this?