Blah I accidentally deleted this the first time I posted it>< And it's not even worth typing out again but I shall!
Okay I got this problem from my calc book. It's easy, but the proof is quite nice. Simple. Yeah so here goes.
Let √2 > a/b, where a, b > 0
Prove (a + 2b)/(a + b) > √2
(a + 2b)/(a + b) > √2
↔ 1 + 1/(a/b + 1) > √2
As a/b > √2,
1 + 1/(a/b + 1) > 1 + 1/(√2 + 1)
(√2  1)(√2 + 1) = 1
√2 = 1 + 1/(√2 + 1)
Thus 1 + 1/(a/b + 1) > √2
and a+2b/a+b > √2 > a/b
