 Solution

At each transition, choose a coordinate uniformly and independently of the previous coordinate choices. Draw a fresh value from that coordinate's full conditional distribution given the current remaining coordinates, and keep all other coordinates fixed. This supplies the random-selection and conditional-draw operations lost in the TeX transcription. The update depends only on the current vector and fresh randomness, which proves the Markov property. The printed mathematical requests are the kernel and its detailed balance, answered in the following two subsections.