If my grammar does not have productions of the form $A\rightarrow\lambda$ and $A\rightarrow B$ for some variables $A$ and $B$ then I know that each step in the derivation must involve an increase in the sentential length. Therefore if I do an exhaustive search for some word $w$ then I know I can stop searching after $|w|$ derivations.
My question is why then do textbooks assert that we can stop after $2|w|$ derivations instead? Yes, I agree that this is true, but wouldn't it be easier just to stop at $|w|+1$ and above? What am I missing for this condition?