A question on placing an upper bound on the probability of a standard brownian motion exiting an interval in a given time interval

Question

I'm supposed to show that there exists a $c > 0 $ such that, for each $P\left(|B_t| \leq \varepsilon, \forall t \in [0, 1]\right) \leq e^{-\frac{c}{{\varepsilon}^2}}$. We're given the hint: $B_{k\varepsilon^2} - B_{(k-1)\varepsilon^2} \quad \text{for } k = 1, \ldots, \left\lfloor \frac{1}{\varepsilon^2} \right\rfloor$ however frankly I'm just completely lost. I understand the hint will give me a sum of independent intervals that sums up to $B_t$, however, I have no idea how I'm supposed to use this. In my mind, I need to use the reflection principle here, but I'm not sure how that's supposewd to interact with the hint I'm given, and I've looked at just reexpressing the condition by looking at the squares of $B_t$ and $\varepsilon$ but it doesn't seem I can apply the reflection principle then, so I'm struggling a bit here. Any help would be appreciated.