Suppose we have a first order language $L$ with interpretation $M$ and domain $D$. Then a denumerable sequence $s=(s_1,s_2,...)$, where $s_j\in D$ for $j=1,2,...$, satisfies a well formed formula $B$ with $x_1,...,x_k$ as free variables if the $k$-tuple $(s_1,...s_k)$ satisfies $B$. This makes perfect sense however, I misunderstand the satisfiability of a well formed formula of the kind $(\forall x_i)B$ where $x_i$ is free in $B$. In Mendelson's "Introduction to Mathematical Logic", he defines the satisfiability of such a wf by stating:
$s$ satisfies $(∀x_i )B$ if and only if every sequence that differs from s in at the most $i$th component satisfies $B$
Does this mean that every sequence $s$ that differs up to the $i$th component satisfies $B$, in other words, every sequence $s$ whose components preceding the $i$th component differ,
or,
every sequence that differs in only the $i$th component satisfies $B$.
The latter seems more of an intuitive definition. A simple clarification would mean a great deal, thank you!