How does cycling happen in the simplex method?
Cycling happens in a different scenario: when $cx_{k+1} = cx_k$. This happens when the current solution is degenerate, that is, when the set $C_k$ is not unique.

For a worked out example, see Cycling in linear programming problems or The simplest examples where the simplex method cycles and conditions where EXPAND fails to prevent cycling. There are many other resources which are available online.