I want some examples that can best illustrate the idea/power/funny of the coupling argument in probability. I think arguments with coupling makes one think in a more probabulistic way. I have a short list of simple examples, but of course there must be more and more.

To number a few examples I can recall:

Proof of the convergence to stationary measure in Markov chain theory, this is now the "classical" way followed by most text books.

Dynkin's card trick. See page 312 on Durrett's book "probability: theory and examples".

Percolation theory: every edge in the lattice graph $\mathbb{Z}^2$ is connected with probability $p$ or disconnected with parobability $1-p$ (independently). Then the measure of the event {there is a path from the origin to infinity} is an increasing function of $p$.

My question is: are there more examples?