If $s_1 +s_2 \gt 1$ and $(t_1,t_2)$ be a convex combination of this with $(0.5,0.5)$ then show that $t_1t_2 \gt 0.25$

Question

Let $(s_1,s_2)$ be such that $s_1 + s_2 \gt 1$.
Let $(t_1,t_2)=((1-\epsilon )(0.5) + \epsilon s_1 , (1- \epsilon)(0.5) +\epsilon s_2)$, where $0< \epsilon \lt1$.

I need to show that for $\epsilon $ small enough $t_1t_2 \gt 0.25$

This is a small portion of a larger proof. I am stuck at this one.

Answer 1

\begin{align}t_1t_2&=(0.5\cdot(1-\epsilon)+\epsilon s_1)\cdot(0.5\cdot(1-\epsilon)+\epsilon s_2)\\&=0.25(1-\epsilon)^2+0.5\cdot\epsilon(1-\epsilon)(s_1+s_2)+\epsilon^2s_1s_2\\ &>0.25(1-\epsilon)^2+0.5\cdot\epsilon(1-\epsilon)+\epsilon^2s_1s_2\\& =0.25-0.5\epsilon+0.25\epsilon^2+0.5\epsilon-0.5\epsilon^2+\epsilon^2s_1s_2\\& =0.25-0.25\epsilon^2+\epsilon^2s_1s_2 \end{align} For small enough $\epsilon$ the last two terms above will be insignificant and therefore $$t_1t_2>0.25$$