Why is this argument used so often by death penalty supporters?

The smart thing to do is actually keep the offenders in for life without parole rather than executing them because it costs much more to execute someone than it is to house them for life. Not many people realize this, it costs millions of dollars to go trial after trial after appeal after appeal etc. before you can actually execute someone. Weird it's so complicated to kill someone right? Jk but really I just see it as a logical thing and try not to get too intimate with it because if I did I would think differently about many things.