The process $\{S_t\}_{t\ge0}$ following $dS_t = \sigma S_tdW_t$ with $S_0>0$ has the solution $$S_t=S_0 e^{-\frac12\sigma^2t+\sigma W_t}$$ Now for any $\epsilon>0$ we have $$\mathbb P(S_t<\epsilon) = \mathbb P\left(Z<\frac{\log(\epsilon/S_0)+\frac12 \sigma^2t}{\sigma\sqrt t}\right) = \Phi\left(\frac{\log(\epsilon/S_0)+\frac12 \sigma^2t}{\sigma\sqrt t}\right)\to1$$ as $t\to \infty$, where $Z\sim N(0,1)$ and $\Phi$ is the CDF of $Z$. So $S_t\to0$ almost surely. But on the other hand $S_t>0$ almost surely for any $t\ge0$.
So my question is, how do these two properties correspond to each other and how is it (intuitively speaking) possible for those two properties to be true at the same time?
Consider $f(t)=\mathrm e^{-t^2}$ for every real $t$. Then $f(t)\gt0$ for every $t$ and $f(t)\to0$ when $t\to\infty$. What is surprising about "th[e]se two properties" [being] "true at the same time"?