# Why does ε-greedy $Q$-learning not oscillate?

I have a intuitive question on the convergence of $Q$- learning. In $Q$ learning for each step a $Q$- value is learned for the state-action pair where the action is selected according to the $\epsilon$-greedy policy determined by $Q$ values.

Now my question is due to $\epsilon$-greedy policy (i.e. due to exploration) is it not possible that the $Q$ values oscillates and does not converge ? each time I am giving chance to non-greedy action his value is improved and may be higher than the value of greedy action after some steps and this becomes greedy now and the same thing continues. Is choosing a small $\epsilon$ enough to prevent this ?