State definitions.
If you have to prove that a certain set is open, begin by writing the definition of
an “open set”.
Perhaps rewrite the definition in the context of the current problem For example, the
textbook definition may say when a set $E$ is open. If you have to show that a set
called $V$ (or something fancy like $\mathfrak F$) is open, then write
According to the definition, $V$ is open if (… here definition of
‘open’ with $E$ replaced by $V$…)
or
According to the definition, $\mathfrak F$ is open if (… here definition of
‘open’ with $E$ replaced by $\mathfrak F$…)
For all, for some.
The theory of metric spaces contains very few long computations like the ones you would
find in calculus. Instead, the place of equations is taken by statements of the form
For every $x$ we have
(some statement involving $x$)
or
For some $x$ we have
(some statement involving $x$)
Synonyms:
-
for all $x$: for every $x$, for any $x$, $\forall x$
-
for some $x$: there is an $x$, there exists an $x$, $\exists x$
Once a variable has been introduced, you can use it.
FAQ.
When we can "choose" $\epsilon$ other than when using contradiction to show that a
point is not the limit of a sequence?
The definition of convergence of a sequence says
“For every $\epsilon\gt0$ there is an $N_\epsilon\in\mathbb N$ such that
… ”
If you are in the situation where you know that a sequence converges (because it is
given, or because you have already proved that yourself), and where you want to
prove something else about the sequence, then you can use the definition: in this
case you get to choose the $\epsilon$, and you can assume that “there is an
$N_\epsilon$ such that … ”
This situation also occurs when you are trying to prove
that the sequence does not converge, and have decided to try a proof by
contradiction. Such a proof begins by saying “Suppose the series
converges”: you
can choose $\epsilon$ as you like, and try to reach a contradiction.
The other kind of situation is that where you don't know that the series converges,
and where you are trying to prove that it converges. In that case you have to
compute or prove the existence of $N_\epsilon$ for every $\epsilon\gt 0$: you do
not get to choose $\epsilon$, and you don't get to assume anything about its value.