(i) Is it possible to have $\phi(n) \ge n$?
Give reasons for your answer.
(ii) Show that $0 < \frac{\phi (n)}{n} < 1$ for $n > 1$.
(i) $\phi(n)$ is defined to be the quantity of natural numbers from 1 to $n$, inclusive, that are coprime to $n$. Since there are only $n$ natural numbers in this range, those coprime are less than or equal to $n$.
However, the natural number 1 is coprime to $n$, and so that quantity, $\phi(n)$, is less than or equal to $(n-1)$, or equivalently, less than $n$.
$$ \phi(n) < n$$
This is equivalent to $\phi(n) \ge n$ being false.
(ii) Since 1 is coprime to any $n>1$ meaning $\phi(n)>0$, and since we're established $ \phi(n) < n$, then
$$ 0 < \phi(n) < n $$
Equivalently
$$ 0 < \frac{\phi (n)}{n} < 1 $$