It is (well-)known that 1 + 2 + 3 + ... + n = n(n + 1)/2.

Since (n + 1)/2 = n/2 + 1/2 <= n for n = 1, 2, 3, ....

We can replace (n + 1)/2 in the above equality with (just) "n", and change to inequality.

If you don't "know" the result, it's a common elementary proof by mathematical induction.