Mean-square criterion for convergence in density ($\frac{1}{N}{\sum }_{n<N}|a_n-\ell {|}^2\to 0$)

The displayed condition implies convergence in density of a sequence to $\ell$, since the fraction of indices with $|a_n-\ell |\ge \epsilon$ is at most ${\epsilon }^{-2}$ times the averaged squared error. A signed Cesaro limit alone is insufficient: $a_n=(-1{)}^n$ has zero signed average but no density limit.