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We remark that if $X$ is an infinite dimensional Banach space then everyseminormalized weakly null sequence in $X$ has an asymptotic monotone basicsubsequence. We also observe that if $X$ contains an isomorphic copy of$\ell_1$, then for every $\varepsilon>0$ there exist a $(1+\varepsilon)$-equivalent norm $\vertiii{\cdot}$ on $X$ such that the unitsphere $(S_{(X, \vertiii{\cdot})})$ contains a normalized bimonotone basicsequences which is symmetrically $2$-separated.

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