Show that for any two positive real numbers $a, b: \frac a{a + 2b} + \frac b{b + 2a} ≥ \frac12$ Question:

Show that for any two positive real numbers $a, b: \frac a{a + 2b} + \frac b{b + 2a} ≥ \frac12$.

So for this question, I began by expanding all terms and moving them all to one side. However, I do not know how to definitively say that the statement is proved.
This is my "work" so far:
$a, b: \frac a{a + 2b} + \frac b{b + 2a} ≥ \frac12$
$\frac {2a(b + 2a) + 2b(a + 2b) - 1(b + 2a)(a + 2b)}{2(a + 2b)(b+2a)} ≥ 0$
$\frac {4ab + 4a^2 + 4b^2 - 2b^2 - 5ab - 2a^2}{4b^2 + 10ab + 4a^2} ≥ 0$
$\frac {2a^2 - ab + 2b^2}{4b^2 + 10ab + 4a^2} ≥ 0$

You almost have it, just use
$$2a^2-ab+2b^2 = 2(a-b)^2+3ab > 0$$
(and the denominator is positive as well).
