Proportionality confusion on Gaussian function (Bayesian Inference)

Question

I am studying this paper here (page 2, the 1st line) -> http://www.stats.ox.ac.uk/~reinert/stattheory/revision09part2.pdf

She is doing some smart things i guess but my level is not that high. In particular, where she derives $\pi{(\beta)} \propto exp(...)$ I don't understand why she doesn't include the remaining term if she had expanded $\frac{-(\beta -\mu)^2}{2\alpha^2}$ correctly.

Answer 1

That's the definition of $\propto$. The remaining term doesn't depend on $\beta$.