# Matching Upper Bound for Online Problems

I have an online problem and I want to prove that no deterministic online algorithm is competitive.

Let $$\texttt{ON}$$ be an online algorithm and $$\texttt{OPT}$$ an optimal offline algorithm (an adversary). Given a sequence of $$n$$ inputs $$\sigma=\sigma_1\sigma_2\cdots\sigma_n$$ revealed in an online fashion, let $$v(\texttt{ON}_\sigma)$$ and $$v(\texttt{OPT}_\sigma)$$ be the values achieved respectively by $$\texttt{ON}$$ and $$\texttt{OPT}$$.

What I proved is this: let $$\sigma=\sigma_1\sigma_2$$ be an 2-input sequence. If $$\texttt{ON}$$ acts on $$\sigma_1$$ with action $$a$$, then I could find an input $$\sigma_2$$ such that $$\frac{v(\texttt{OPT}_\sigma)}{v(\texttt{ON}_\sigma)},$$ is unbounded.

The problem is that, if the action $$a$$ changes, then the above ratio could be equal to $$1$$.

Is it sufficient to prove that the ratio is unbounded for some action $$a$$? Or must it be true for any $$a$$?