sequential characterization of boundedness

Theorem [1, 2]A set $B$ in a real (or possibly complex)topological vector space $V$is bounded (http://planetmath.org/BoundedSetInATopologicalVectorSpace) if and only if thefollowing condition holds:

1. If ${\{z_i\}}_{i=1}^{\mathrm{\infty }}$ is a sequence in $B$, and ${\{{\lambda }_i\}}_{i=1}^{\mathrm{\infty }}$is a sequence of scalars (in $ℝ$ or $ℂ$),such that ${\lambda }_i\to 0$, then ${\lambda }_i{z}_i\to 0$ in $V$.