# 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 $$\liminf_{n\rightarrow \infty} a_n = 0$$.