Convergence by synchronous coupling for bounded continuous test functions

If two coupled processes approach each other in L2 and one family of marginals is uniformly tight, then their expectations of any bounded continuous test function approach each other. On a large compact set use uniform continuity; outside it use boundedness; control excessive coupling separation by the second-moment Markov bound. A constant expectation for one process then identifies the other limit. No global derivative or Lipschitz bound is necessary.