On, Alan F. Beardon's, 'Limits: A New Approach to Real Analysis' it's asked:
Show (i) by induction, and (ii) by using the Least Upper Bound Axiom, that any set of n real numbers contains a largest element. Show also that it contains a smallest element.
I'm not concerned with (i); but I've been struggling with (ii).
I've sketched:
Take $a = 1 + \sum_{i=0}^n |x_i|$; and any $x_j$.
Well: $a = |x_j| + (1 + \sum_{i \in ([1..n] - \lbrace j\rbrace)} |x_i|)$
So: $\forall i \in [1..n]. a > x_i$, ie $x_{[1..n]}$ has an upper bound.
Therefore $x_{[1..n]}$ has a supremum, by the 'LUB Axiom', $\sup x_{[1..n]} = s$.
Suppose $s \notin x_{[1..n]}$; and so, consider $\epsilon_i$ such that $x_i = s - \epsilon_i$, $\epsilon_i \gt 0$.
Similarly, $\epsilon_{[1..n]}$ has an infimum, by the 'GLB Theorem', $\inf \epsilon_{[1..n]} = \epsilon$.
Take $ b_i = \begin{cases} 1/2 & \epsilon_i = 1 \\ 1/\epsilon_i & \epsilon_i \gt 1 \\ (\epsilon_i)^2 & \epsilon_i \lt 1 \end{cases} $
So $b_i \lt \epsilon_i$ and $b_i \lt 1$.
Now take $b = \prod_{i}^n b_i$; and any $b_j$.
Well: $b = b_j\prod_{i \in ([1..n] - \lbrace j\rbrace)} b_i$.
Since $\prod_{i \in ([1..n] - \lbrace j\rbrace)} b_i \lt 1$, we have $\epsilon_j > b_j > b$.
So $\epsilon > 0$, since $b$ is a lower bound of $\epsilon_{[1..n]}$ and $b > 0$.
And we have $x_i \leq s - \epsilon$, ie $s - \epsilon$ is an upper bound of $x_{[1..n]}$, contradicting the minimality of $s$, since $s - \epsilon < s$.
$\therefore s \in x_{[1..n]}$, ie $x_{[1..n]}$ has a biggest element.
However, I might have used Induction in the sketch above, especially while managing:
$a = 1 + \sum_{i=0}^n |x_i|$ and $b = \prod_{i}^n b_i$.
Is it the case that I've used induction in my sketch above?
To me it seems that whenever you have an $A \subsetneq \mathbb{N}$ you can get by without induction; eg in 'Finite Dimensional Linear Algebra', or the question above.
As in:
https://math.stackexchange.com/a/1311896/940301
https://math.stackexchange.com/a/626648/940301
How would you go about proving (ii), from first principles, without Induction [and without the Axiom of Choice]?
