A visual intuitive proof that **√ab** cannot be larger than **(a+b)/2**, where a, b ∈ R*+

You can also prove it just by using properties of numbers:

( a – b )² ≥ 0 ⇔ a² – 2ab + b² ≥ 0 ⇔ a² + b² ≥ 2ab ⇔ a² + 2ab + b² ≥ 4ab

⇔ ( a + b )² ≥ 4ab ⇔ **√ab ≤ ( a+b )/2**