Let $f _ { 1 }$ be a positive constant function on $[ 0,1 ]$ with $f _ { 1 } ( x ) = c$, and let $p$ and $q$ be positive real numbers with $1 / p + 1 / q = 1$. Moreover, let $\left\{ f _ { n } \right\}$ be the sequence of functions on $[ 0,1 ]$ defined by
$$f _ { n + 1 } ( x ) = p \int _ { 0 } ^ { x } \left( f _ { n } ( t ) \right) ^ { 1 / q } \mathrm {~d} t \quad ( n = 1,2 , \ldots )$$
Answer the following questions.
(1) Let $\left\{ a _ { n } \right\}$ and $\left\{ c _ { n } \right\}$ be the sequences of real numbers defined by $a _ { 1 } = 0 , c _ { 1 } = c$ and
$$\begin{aligned} & a _ { n + 1 } = q ^ { - 1 } a _ { n } + 1 \quad ( n = 1,2 , \ldots ) \\ & c _ { n + 1 } = \frac { p \left( c _ { n } \right) ^ { 1 / q } } { a _ { n + 1 } } \quad ( n = 1,2 , \ldots ) \end{aligned}$$
Show that $f _ { n } ( x ) = c _ { n } x ^ { a _ { n } }$.
(2) Let $g _ { n }$ be the function on $[ 0,1 ]$ defined by $g _ { n } ( x ) = x ^ { a _ { n } } - x ^ { p }$ for $n \geq 2$. Noting that $a _ { n } \geq 1$ holds true for $n \geq 2$, show that $g _ { n }$ attains its maximum at a point $x = x _ { n }$, and find the value of $x _ { n }$.
(3) Show that $\lim _ { n \rightarrow \infty } g _ { n } ( x ) = 0$ for any $x \in [ 0,1 ]$.
(4) Let $d _ { n }$ be defined by $d _ { n } = \left( c _ { n } \right) ^ { q ^ { n } }$. Show that $d _ { n + 1 } / d _ { n }$ converges to a finite positive value as $n \rightarrow \infty$. You may use the fact that $\lim _ { t \rightarrow \infty } ( 1 - 1 / t ) ^ { t } = 1 / \mathrm { e }$.
(5) Find the value of $\lim _ { n \rightarrow \infty } c _ { n }$.
(6) Show that $\lim _ { n \rightarrow \infty } f _ { n } ( x ) = x ^ { p }$ for any $x \in [ 0,1 ]$.