Assume that $(x_{n})$ is a generic sequence of points such that $h_{\overline{L}}(x_{n})$ tends to $0$. By Theorem 3.3.1, $h_{\overline{M}}(x_{n})$ converges to

$$\frac{1}{\ell}({\widehat{c}}_{1}(\overline{L}){\widehat{c}}_{1}(\overline{M})|X).$$

(3.4.2)

Except when both lower bounds are zero, this is strictly bigger than the lower bound of the proposition, which is equal to

$$\frac{1}{\ell+m}({\widehat{c}}_{1}(\overline{L}){\widehat{c}}_{1}(\overline{M})|X).$$

In other words, the greedy obvious method to find points of small height for $\overline{L}+\overline{M}$ that first minimizes the height $h_{\overline{L}}$, only works up to the factor $(\ell+m)/\ell>1$.