Every probability vector is approximated by n-types

Fix a listing $a_1,\dots ,a_k$ of $\mathcal{A}$. Set $m_i=\lfloor nq(a_i)\rfloor$ and $r=n-{\sum }_i{m}_i={\sum }_i(nq(a_i)-m_i)$. Thus $r$ is an integer with $0\le r<k$. If $r>0$, at least $r$ fractional parts are positive, since fewer than $r$ positive terms, each below $1$, cannot sum to $r$. Add $1$ to the first $r$ positive fractional-part coordinates in the fixed listing; if $r=0$, add nothing. The resulting integers $m_i^{\mathrm{\prime }}$ are nonnegative and sum to $n$. If $k=1$, then $q(a_1)=1$ and $r=0$, so this gives the unique type exactly.

Put $q_n(a_i)=m_i^{\mathrm{\prime }}\mathrm{/}n$. This is an $n$-type by [F1]. Each coordinate changes from $q(a_i)$ by less than $1\mathrm{/}n$: it is rounded down when not selected and rounded up when selected. Summing the $k$ coordinate errors gives $\mathrm{\parallel }{q}_n-q{\mathrm{\parallel }}_1<k\mathrm{/}n$, hence the stated weak bound. The construction also applies when some $q(a_i)=0$.