A note on asymptotically monotone basic sequences and well-separated sets

We remark that if $X$ is an infinite dimensional Banach space then every seminormalized weakly null sequence in $X$ has an asymptotic monotone basic subsequence. We also observe that if $X$ contains an isomorphic copy of ${\mathrm{\ell }}_1$, then for every $\epsilon >0$ there exist a $(1 +\varepsilon)$-equivalentnorm$\vertiii{\cdot}$on$X$ such that the unit sphere $(S_{(X,\text{vertiii}\cdot )})$ contains a normalized bimonotone basic sequences which is symmetrically $2$-separated.