Show that if $M^n$ is smooth a manifold and $A,B$ are closed disjoint sets on $M$ then there is a smooth function $ 0\le f \le 1$ with $f(A) = 0$ and $f(B) = 1.$
What I am trying is:
Note that $X-A$ is open, as well $X-B.$ In particular, $X - A \cap X - B = \emptyset$ and $X - A \cup X - B = X.$
Lets define $U = X - A$ and $V = X - B.$ Then there is a unity partition subordinated to this cover. By properties of unity partition, $\{f_U,f_V\}$, $f_U(A) = 0$ and $f_V(B) = 0.$ Even more, $f_A$ and $f_B$ are smooth. So, we can then define $g(x) = \frac{f_U(x)}{f_V(x) + f_U(x)}$. Then this is the searched function. Indeed, $0 \le g(x) \le 1$ and $g(A) = 0$, $g(B) = 1.$
Is this right?