Volcano

$f_{n}(x)$ = $x^{n} + x^{n-1} + ... + x - 1$ is a continuous function. $f_{n}(0) = -1$ and $f_{n}(1) = n -1$ $\Rightarrow$ $f_n$ has at least one real positive root $a_n$ due to Bolzano's theorem (intermediate value theorem for continuous functions). Furthemore, the function is strictly increasing for $x\ge 0$ due to its derivate is positive for $x\ge 0$ $\Rightarrow$ this root $a_n$ is the only positive real root.