Set of real numbers, bounds and sequences I

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Let $A$ be a subset of $\R$. Then a real number $y$ is an upper bound for $A$ if $A \subseteq (-\infty,y]$, i.e., for all $a \in A$ we have $a\leq y$. We define lower bound in a similar way.

Not every set has an upper bound and/or a lower bound (in $\R$). For example $\R$, $\Q$ and $\Z$ have no upper bounds and they also have no lower bounds (in $\R$). Other sets such as $\N$, $(-\infty,2)$, $[-3,5)$ have one or the other or both. 

The empty set $\emptyset$ is a pathological example: every real number is both an upper bound and a lower bound for $\emptyset$, so in particular many upper bounds for $\emptyset$ are less than some of the lower bounds for $\emptyset$. Let's face it, $\emptyset$ always causes trouble!

The connection between English and mathematics is not always straightforward and can lead to misunderstandings!

We say that the subset $A$ of $\R$ is bounded above if there is at least one upper bound for $A$ in $\R$. Of course as soon as $A$ has an upper bound $y\in\R$, then $y+1$, $y+0.23$ and generally anything $>y$ is also an upper bound for $A$. So $A$ will then have infinitely many upper bounds.

We define bounded below similarly. If  $y\in\R$ is a lower bound for $A$, then so is everything $<y$ (just as with upper bounds).

If $A$ is not bounded above then $A$ is unbounded above. Similarly, if $A$ is not bounded below then $A$ is unbounded below.

The set $A$ is bounded if it is both bounded above and bounded below (in which case there are real numbers $y_1$ and $y_2$ with $A \subseteq [y_1,y_2]$).

The set $A$ is unbounded if it is not bounded, i.e., $A$ is unbounded above or $A$ is unbounded below (or both: this is the mathematician's 'or').

Warning!

Note that $A$ is bounded if and only if it is both bounded below and bounded above. But $A$ is unbounded if $A$ is unbounded below or unbounded above (or both). From the English alone you might not realise which was 'and' and which was 'or'. So probably it is best to make sure that you have thought about lots of sets and which definitions do and don't apply to them.

I think that I should make a video called Think of a set to go along with with my Think of a function video (and another video that could be called Think of a sequence) in our old Core Topics in University Mathematics videos on YouTube.

Now let $A$ be a non-empty subset of $\R$, and suppose that $A$ is bounded above. Our completeness axiom for $\R$ tells us that $A$ has a least upper bound (or supremum) $S=\sup A \in \R$. We can then say that the set of all upper bounds (in $\R$) for $A$ is precisely the interval $[S,\infty)$.

Similarly, if $A$ is a non-empty subset of $\R$ which is bounded below, then $A$ has a greatest lower bound (or infimum) $s=\inf A \in \R$. We can then say that the set of all lower bounds for $A$ (in $\R$) is precisely the interval $(-\infty,s]$.

Exercise: Investigate the sets of all upper bounds and the sets of all lower bounds for the various specific examples of sets mentioned above, including the unbounded sets and the empty set.

In the next part we will look at the connections between sets of real numbers, bounds, supremum, infimum, sequences and epsilons!

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