= Solution
The update defines a time-homogeneous <Markov process>: given its present value, the next value depends only on independent fresh input. Part (b)(i) identifies an invariant law: initializing with the rate-one <exponential distribution> preserves that distribution at every time. Part (b)(iii) shows convergence of the one-time laws to this invariant law from every deterministic initial value.
For a countable <state space>, the analogous <countable-state Markov chain convergence theorem> says that an <irreducible Markov chain> that is <aperiodic> and <positive recurrent> converges to its unique <stationary distribution> from every initial state. Here the state space is continuous, so that countable-state theorem does not directly supply the proof; the explicit <coupling> does. The two conclusions have the same equilibrium interpretation.
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