Is my proof correct? I have to prove that:
All positive integers $n$ are equal
$P (n)$: All numbers in a set of $n$ positive integers are equal
Base: $P (1)$ is true
Inductive hypothesis: Suppose that $P(n)$ is true
Induction step: Let $A$ be the set of the first $n + 1$ positive integers.
Let $A'$ the set of the first $n$ positive integers in $A$ (the smallest $n$ integers in $A$).
Let $A''$ be the set of the last $n$ positive integers in $A$ (the largest $n$ integers in $A$).
Then from the fact that $P (n)$ is true we know that:
all integers in $A'$ are equal
all the integers in $A''$ are the same
Thus the first $n$ elements of $A$ are equal and the last $n$ elements of $A$ are equal.
So all the elements of $A$ are the same.
So all positive integers are equal.