How Splitting Summation method works
$$\begin{align*}
55 = 1 + 2 + \dots + 10 &amp;= (1 + 2 + 3 + 4 + 5) + (6 + 7 + 8 + 9 + 10)\\
&amp;\color{red}{\geq} (0 + 0 + 0 + 0 + 0) + (5 + 5 + 5 + 5 + 5) \\
&amp;= (10/2)(10/2) = 25\,.\end{align*}$$

If you're wondering how to come up with such substitutions, there's no general method: this is one of the creative things you have to do to produce mathematical proofs.