M be a TM accepting L. Exists since looking at properties of recognizable langs. Let L' be a finite subset of L with property P. Exists by second assumption. Let j be max number of steps by M to accept a word in L'. Previously shown recognizable by construction. Has containment property.
Finite sets are enumerable.
Enumerating TM list all finite sets, in increasing size. When recognizable, unclear if Rice's Thm or direct construction easier. Previously saw that it fails the second assumption. When L is unrecognizable, usually one of first two conditions fail.