Lattice precoding is an effective strategy for multiantenna broadcast. In this paper, we show that approximate lattice precoding in multiantenna broadcast is a variant of the closest vector problem (CVP) known as $η$-CVP. The proximity factors of lattice reduction-aided precoding are defined, and their bounds are derived, which measure the worst-case loss in power efficiency compared to sphere precoding. Unlike decoding applications, this analysis does not suffer from the boundary effect of a finite constellation, since the underlying lattice in multiantenna broadcast is indeed infinite.