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In this note, we investigate the convergence of a $U$-statistic of order two having stationary ergodic data. We will find sufficient conditions for the almost sure and $L^1$ convergence and present some counter-examples showing that the $U$-statistic itself might fail to converge: centering is needed as well as finiteness of $\sup _{j\ge 2}\mathbb{E}[|h(X_1,X_j)|]$.