  Let $F$ be an infinite field and let $f$ be a nonzero multilinear polynomial
with coefficients in $F$. We prove that for every positive integer $d$ there
exists a positive integer $s$ such that $f(M_{s}(F))$, the image of $f$ in
$M_{s}(F)$, contains all trace zero $d \times d$ matrices. In particular, the
image of $f$ in the algebra of all finitary matrices contains all trace zero
finitary matrices.
