= Finite CW approximation from bounded homology
{title2=$H_i(X)=0\ (i>n)\Rightarrow\dim C\leq n+1$}
Start with a finite $n$-dimensional $n$-connected approximation. Its degree-$n$ <homology> kernel is finite free. Realize a basis via the <Relative Hurewicz theorem> and attach $(n+1)$-cells; their boundary is injective, so they kill precisely that kernel without adding top <homology>. The <homological Whitehead theorem> gives a finite weak equivalence of dimension at most $n+1$.
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