Let $(X_1,X_2,...)$ be i.i.d random variables with mean $0$ and variance $1$. By the Law of the Iterated Logarithm, for all $\epsilon >0$,
\begin{equation} P\left[ \frac{1}{t}\sum_{i=1}^{t}X_i \geq (1+\epsilon)\sqrt{\frac{2\log \log t}{t}}\text{ i.o.}\right] =0 \end{equation}
I want to show that \begin{equation} P\left(\frac{1}{t}\sum_{i=1}^{t}X_i<(1+\epsilon)\sqrt{\frac{2\log \log t}{t}}\text{ for all }t \geq 3 , \;\epsilon >0\right) >0 \end{equation} i.e. with positive probability, $\frac{1}{t}\sum_{i=1}^{t-1}X_i$ never exceeds the LIL bound.