PomeLemma 8.6.4.EBF9BA687F39 Every bounded upper semicontinuous real function $u$ on $X$ is the pointwise decreasing limit of smooth functions $h_ j>u$.
Lemma 8.6.4EBF9BA687F39
Proof. Fix a Riemannian distance $d$ on $X$ and put
\[ c_ j(x)=\sup _{y\in X}\{ u(y)-j d(x,y)\} . \]
Each $c_ j$ is Lipschitz, $c_ j\geq u$, and $c_{j+1}\leq c_ j$. The supremum is attained. If $y_ j$ attains it at a fixed $x$, then $j d(x,y_ j)\leq \sup _ X u-u(x)$, so $y_ j\to x$. Upper semicontinuity gives $\limsup _ j c_ j(x)\leq u(x)$, and hence $c_ j\downarrow u$ pointwise.
Set $\delta _ j=4^{-j}$. Smooth approximation of continuous functions on a compact manifold gives $s_ j\in C^\infty (X)$ with $\| s_ j-c_ j\| _\infty \leq \delta _ j/4$. Take $h_ j=s_ j+\delta _ j$. Then
\[ u+\tfrac 34\delta _ j\leq h_ j\leq c_ j+\tfrac 54\delta _ j, \qquad h_{j+1}\leq c_ j+\tfrac 5{16}\delta _ j < c_ j+\tfrac 34\delta _ j\leq h_ j. \]
The bounds also imply $h_ j(x)\to u(x)$ at every point. $\square$
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