In Lee's "Introduction to Smooth Manifolds" 2nd edition page 43 he gives a proof for the existence of partitions of unity for smooth manifolds without boundary and there is one part of the proof I would like clarification of.
So from what I understand we have already established the existence of a smooth bump function (Lemma 2.22) $H(x) = h(|x|): \mathbf{R}^n \to \mathbf{R}$ such that for $0< r_1< r_2$, $r_1, r_2 \in \mathbf{R}$, $H \equiv 1$ on $\overline{B_{r_1}}(0)$, $0<H<1$ on $B_{r_2}(0) \setminus \overline{B_{r_1}}(0)$ and $H \equiv 0$ on $\mathbf{R}^n \setminus B_{r_2}(0)$.
Now this leads me to believe that in constructing the partitions of unity for a smooth manifold that we would like to use a similar construction for a covering of our manifold $M$ by regular coordinate balls (A coordinate ball $B\subset M$ is regular if there exists positive real numbers $r < r'$ and a smooth coordinate ball $B'$ such that $\overline{B} \subset B'$ and there is a smooth coordinate map $\varphi: B' \to \mathbf{R}^n$ such that $\varphi(B) = B_r(0)$, $\varphi(\overline{B}) = \overline{B}_r(0)$, and $\varphi(B')= B_{r'}(0)$). By this I mean that we construct our bump function for each regular coordinate ball so that it is $1$ on $\overline{B}_r(0)$, transitions to $0$ on $B_{r'}(0) \setminus \overline{B}_r(0)$ and is zero outside of $B_{r'}(0)$.
Now for the proof: Lee picks a countable basis of regular coordinate balls $\{B_i\}$ on the manifold, which has a locally finite refinement. Then since each coordinate ball is regular we have $r_i< r_i'$ and $B_i'$ such that $B_i \subset B_i'$ a smooth coordinate chart $\varphi_i:B_i' \to \mathbf{R}^n$ as defined above. Thus for each $i$ indexing the regular coordinate balls we define the following function on our manifold $f_i: M \to \mathbf{R}$ as:
$f_i(x) = H_i \circ \varphi_i$ for $x \in B_i'$ where $H_i$ is the bump function defined with respect to $r_i, r_i'$.
$f_i(x) = 0$ for $x \in (M \setminus \overline{B}_i)$
Lee then states the following:
"$H_i$ is a smooth function which is positive in $B_{r_i}(0)$ and zero outside of it."
From the previous establishment of the smooth bump functions this leads me to believe that our bump function should have been chosen so that it is $1$ on $\overline{B}_{r_i}(0)$, then transition to zero on $B_{r_i'}(0) \setminus \overline{B}_{r_i}(0)$, and be zero outside of it.
What am I missing here?
Thanks in advance for your help.
Notice that all I needed in the proof of Theorem 2.23 was that $H_i$ is positive in $B_{r_i}(0)$ and zero elsewhere -- not that it's $1$ on $\overline B_{r_i}(0)$ and zero outside of $B_{r_i'}(0)$. To apply Lemma 2.22 in this situation, you want to take $r_2$ in the lemma to be $r_i$, and $r_1$ in the lemma to be some positive number less than $r_i$.