# Mistake using Borel-Cantelli Lemma

Ah, I see. In that case you are correct that $P(\overline{X}_n < c \text{ e.v.})=0$, but this is irrelevant, since we may have $\liminf_{n\rightarrow \infty} \overline{X}_n < c$ even if we do not have $\overline{X}_n < c \text{ e.v.}$