Let {X"; n > 0} be a Harris-recurrent Markov chain on a general state space. It is shown that there is a sequence of random times {N¡; i > 1} such that {XN.; i > 1} are independent and identically distributed. This idea is used to show that {Xn} is equivalent to a process having a recurrence point, and to develop a regenerative scheme which leads to simple proofs of the ergodic theorem, existence and uniqueness of stationary measures.