We have trace map as $tr : M_n(F) \to F.$
Then using trace map is linear map I get $||tr(A)-tr(B)||=||tr(A-B)|| \le ||tr||||A-B||.$
Since I know that trace is a continuous map, I have $||tr||$ as Lipschitz constant. (Because $||tr||=\sup_{||A||=1}||tr(A)||$ and as the set $||A||=1$ is compact, the image $tr(A)$ is also compact hence obtaining maximum and minimum.)
By Lipschitz condition, I get trace map is uniformly continuous.
Am I right?