Let $\epsilon ' = \epsilon (s+t - \epsilon)$, where $s+t > 0$, and $\epsilon > 0$ can be made arbitrarily close to 0. How can I prove that $\epsilon '$ can made as close to zero as we like? Here is the way I think about it: We know that $\epsilon ' = \epsilon (s+t - \epsilon) < \epsilon (s+t) = \epsilon a$, where $a = s+t > 0$. Since $\epsilon$ can be made arbitrarily close to zero, $\epsilon a$ can be made arbitrarily close to zero as well, hence $\epsilon '$ can be made arbitrarily close to zero.
Is this correct logic? What would be a more correct or better proof?
That is correct. You are making sure the factor $s+t-\varepsilon$ is bounded for $\varepsilon > 0$, so that when $\varepsilon$ goes to $0$, $\varepsilon'$ goes to $0$ as well.