Bayesian statistics: show that corresponding posterior is a proper distribution
By Bayes's rule, $p(\lambda \mid y)$ is proportional to $p(\lambda) p(y \mid \lambda) \propto \frac{1}{k!} (n\lambda)^k e^{-n \lambda}$ where $k=\sum_{i=1}^n y_i$. Explicitly, the posterior density is $$p(\lambda \mid y) = c (n\lambda)^k e^{-n \lambda}$$ where $c^{-1} = \int_0^\infty (n \lambda)^k e^{-n \lambda} \, d\lambda$. Check that this is the Gamma distribution mentioned in the problem.