= Solution
Choose an initial value $X_0$. One sweep of the <Gibbs sampler> consists of
$$
Y_m\sim f_{Y\mid X}(\,\cdot\mid X_{m-1}),
\qquad
X_m\sim f_{X\mid Y}(\,\cdot\mid Y_m).
$$
After a burn-in period, the pairs $(Y_m,X_m)$ are retained as approximate samples. Standard irreducibility and recurrence conditions are needed for convergence from the chosen starting point.
The transition density from $(y,x)$ to $(y',x')$ is
$$
k((y,x),(y',x'))
=f_{Y\mid X}(y'\mid x)f_{X\mid Y}(x'\mid y').
$$
Suppose $(Y,X)$ currently has density $f_{XY}$. The density after one sweep is
$$
\begin{aligned}
&\int_{\mathbb R^2}
f_{XY}(y,x)
f_{Y\mid X}(y'\mid x)
f_{X\mid Y}(x'\mid y')\,dy\,dx\\
&\quad=f_{X\mid Y}(x'\mid y')
\int_{\mathbb R}f_X(x)f_{Y\mid X}(y'\mid x)\,dx\\
&\quad=f_{X\mid Y}(x'\mid y')f_Y(y')
=f_{XY}(x',y').
\end{aligned}
$$
This proves the <stationarity of the two-coordinate Gibbs sampler>:
$$
\boxed{f_{XY}\text{ is stationary for the Gibbs transition kernel}.}
$$
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