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Convergent Series and lim inf

This little result came up when proving the convergence of a stochastic gradient algorithm and I want to write it down to remember it after discussions with Matt Johnson and Alex Tank.

Let \(a_1, a_2, \ldots\) be a positive sequence of numbers. If \(\sum_{n=1}^\infty \frac{1}{n} a_n < \infty\), then \(\liminf_{n\rightarrow \infty} a_n = 0\).

The proof is by contrapositive. Assume that \(\liminf_{n \rightarrow \infty} = c \neq 0\). Then by the \(\epsilon\) characterization of the limit inferior we have that \(\forall \epsilon > 0\), \(\exists N\) such that for \(n > N\), \(a_n > c - \epsilon\). This implies that

$$ \begin{aligned} \sum_{n=1}^\infty \frac{1}{n} a_n & \geq \sum_{n'=N}^\infty \frac{1}{n'}a_{n'} \\ & > (c - \epsilon) \sum_{n'=N}^\infty \frac{1}{n'} = \infty. \end{aligned} $$

But we assumed that the original sum converged, implying that \(\liminf_{n\rightarrow \infty} a_n = 0\).