I want to show equivalence of following two definitions:
Definition 1: A subset $U$ of a topological vector space is called bounded, if for every neighborhood of $0$ $V$, there is a scalar $s\in\mathbb{R}$, such that $U\subseteq sV$.
Definition 2: A subset $U$ of a topological vector space is called bounded, if for every neighborhood of $0$ $V$, there is a scalar $s'>0$, such that $U\subseteq s'V$.
It is clear to me, that a Definition 2-bounded set is Definition 1-bounded, since any $s'>0$ in that definition is a scalar $s$ in $\mathbb{R}$.
To show that a Definition 1-bounded set is Definition 2-bounded set, is clear to me in the cases, when $s\ge 0$. In the case $s>0$, there is clearly $s'=s>0$. In the case $s=0$, it follows $U\subseteq\{0\}$, i.e. $U\in\{\emptyset, \{0\}\}$. Then one can find $s'=1$, since $\emptyset, \{0\}\subseteq V$.
However, in the case $s<0$ I am not sure, how to show it. The neighborhood of $0$ $V$ does not have to be balanced, so just taking $s'=-s$ does not work.
Any hint or help is appreciated! Thank you in advance!