Tight bound proof

Suppose that for a classifier, the worst-case error rate is shown to be $\leq \frac{1}{4}$. I want to prove that $\frac{1}{4}$ is a tight upper bound.
To do so, my approach is that it is sufficient to find a specific case under which $\frac{1}{4}$ is reached exactly. However, I was only able to show that this error rate probability, as a function of $\epsilon$, tends to $\frac{1}{4}$ as $\epsilon \rightarrow 0$, i.e. $p(\epsilon)\rightarrow \frac{1}{4}$.

My question is whether my approach is correct and also whether this is sufficient as a demonstration of tightness or not. If so, any reference in this regard would be helpful as well.

Thanks a ton!