I was reading this paper (Friedman et al, 2010, Regularization Paths for Generalized Linear Models via Coordinate Descent) describing the coordinate descent algorithm for LASSO, and I can't quite figure out how the soft-thresholding update for each $\beta_j$ is derived for the linear regression case. Specifically, how does one go from equation (4) to equation (5) in the paper? Can anyone show me a detailed derivation of the update? Thanks!

I was reading this paper (Friedman et al, 2010, Regularization Paths for Generalized Linear Models via Coordinate Descent) describing the coordinate descent algorithm for LASSO, and I can't quite figure out how the soft-thresholding update for each $\beta_j$ is derived for the linear regression case. Specifically, how does one go from equation (4) to equation (5) in the paper? Can anyone show me a detailed derivation of the update? Thanks!

Edit: actually this paper deals with the elastic net penalizing term, which reduces to LASSO when $\alpha=1$. I am mostly interested in LASSO, but I guess the derivation should be principally the same for elastic net.