Let $n \in \mathbb { N }$. Show that we can order the zeros of $\varphi _ { n }$, that is, there exists a strictly increasing sequence $\left( \alpha _ { k } ^ { ( n ) } \right) _ { k \in \mathbb { N } }$ of zeros of $\varphi _ { n }$ such that $\varphi _ { n }$ does not vanish on $] 0 , \alpha _ { 0 } ^ { ( n ) } [$ and on every interval $] \alpha _ { k } ^ { ( n ) } , \alpha _ { k + 1 } ^ { ( n ) } [$ with $k$ in $\mathbb { N }$ and that $\lim _ { k \rightarrow \infty } \alpha _ { k } ^ { ( n ) } = + \infty$. Construct the sequence $\left( \alpha _ { k } ^ { ( n ) } \right) _ { k \in \mathbb { N } }$ by induction on $k$ by showing that the set $\mathcal { Z } _ { k }$ of zeros of $\varphi _ { n }$ in the interval $] \alpha _ { k } ^ { ( n ) } , + \infty [$ has a smallest element.
Let $n \in \mathbb { N }$.
Show that we can order the zeros of $\varphi _ { n }$, that is, there exists a strictly increasing sequence $\left( \alpha _ { k } ^ { ( n ) } \right) _ { k \in \mathbb { N } }$ of zeros of $\varphi _ { n }$ such that $\varphi _ { n }$ does not vanish on $] 0 , \alpha _ { 0 } ^ { ( n ) } [$ and on every interval $] \alpha _ { k } ^ { ( n ) } , \alpha _ { k + 1 } ^ { ( n ) } [$ with $k$ in $\mathbb { N }$ and that $\lim _ { k \rightarrow \infty } \alpha _ { k } ^ { ( n ) } = + \infty$.
Construct the sequence $\left( \alpha _ { k } ^ { ( n ) } \right) _ { k \in \mathbb { N } }$ by induction on $k$ by showing that the set $\mathcal { Z } _ { k }$ of zeros of $\varphi _ { n }$ in the interval $] \alpha _ { k } ^ { ( n ) } , + \infty [$ has a smallest element.