# How to show this distribution is proper?

This question is based on problem 14 from chapter 3 of Gelman et al.'s Bayesian Data Analysis.

We have four data points $$y_i$$ with covariates $$x_i$$, $$i=1,\cdots,4$$. We use the model: $$y_i\sim \mathrm{Bin}(n_i,\theta_i),\quad \mathrm{logit}(\theta_i) = \alpha + \beta x_i.$$ We place an improper uniform prior $$(\alpha,\beta)\sim1$$. The problem is to show that the resulting posterior distribution of $$\alpha,\beta|y,x,n$$ is proper over the range $$(\alpha,\beta)\in\mathbb R^2$$.

What I have tried:
The posterior distribution is (proportional to): $$\prod_{i=1}^4 \left[ \left( \frac{\mathrm{exp}(\alpha+\beta x_i)}{1+\mathrm{exp}(\alpha+\beta x_i)} \right)^{y_i} \left( \frac1{1+\mathrm{exp}(\alpha+\beta x_i)}\right)^{n_i-y_i} \right]$$. This is less than or equal to $$\frac1{1+\mathrm{exp}(\alpha+\beta x_1)}$$, and that upper bound is tight, since depending on the values of $$y$$ and $$n$$ the (unnormalized) posterior can reach that upper bound. But surely $$\int\int_{\mathbb R^2}\frac1{1+\mathrm{exp}(\alpha+\beta x_1)}d\alpha d\beta=\infty$$, since for any $$\delta$$ it's easy to see that the integrand is greater than $$1-\delta$$ over an infinite region of $$\mathbb R^2$$.

Where have I gone wrong?