How can I prove (preferrably without the use of any heavy theorems) the existence of a smooth function $\mu\!:\mathbb{R}\rightarrow\!\mathbb{R}$ with properties $\mu(0)\!>\!\varepsilon$, $\:\mu_{[2\varepsilon,\infty)}\!=\!0$, $\:-1\!<\!\mu'\!\leq\!0$?
I'm guessing bump functions should come in handy, but I don't know how to bound the derivative.
This is needed in the proof of Morse handle attachment theorem: Banyaga & Hurtubise, p.65.