Finite CW approximation from bounded homology ($H_i(X)=0(i>n)\Rightarrow \dim C\le n+1$)

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 $n+1$.