Section G.4 Quantifiers are not optional
Many of our definitions and theorems involve an awful lot of \(\forall\)s and \(\exists\)s, either written using those symbols or, just as often, written out in words. These are essential and cannot be dropped.
Consider, for instance Definition 1.29, which says that a sequence \(\seq xn\) in a metric space \((X,d)\) converges to \(x_0 \in X\) if and only if
\begin{align}
\begin{aligned}
\amp \text{for any } \varepsilon \gt 0 \text{ there exits } N \in \N \text{ such that,} \\
\amp \text{for all } n \ge N,\, d(x_n,x_0)\lt\varepsilon.
\end{aligned}\tag{✶}
\end{align}
I occasionally see students trying to abbreviate (✶) as
\begin{gather}
d(x_n,x_0)\lt\varepsilon \quad \forall
\varepsilon \gt 0\text{.}\tag{✶✶}
\end{gather}
This means something completely different! Since \(d\) is a metric, the left hand side \(d(x_n,x_0)\) is a non-negative real number. If it is really less than \(\varepsilon\) for any \(\varepsilon
\gt 0\text{,}\) then the only possibility is \(d(x_n,x_0)=0\text{.}\) Since \(d\) is a metric, this then forces \(x_n=x_0\text{.}\) In summary, by taking the statement \(x_n \to x_0\) in (✶) and dropping some of the quantifiers, we have arrived at the altogether different statement (✶✶) that \(\seq xn\) is the constant sequence \((x_0,x_0,\ldots)\text{.}\)
So how can we avoid having to write complicated sentences with tons of quantifiers? Often we cannot! On the other hand, in many situations things simplify quite a bit if we use the appropriate results and jargon from lectures — indeed, this is part of why we introduced this jargon and proved these results in the first place.
