Limit of nth Root of Sum of nth Powers equals Maximum

Theorem
Let $a_1, a_2, \ldots, a_r$ be non-negative real numbers.

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $a = \max \set {a_1, a_2, \ldots, a_r}$.

Then:
 * $\ds \lim_{n \mathop \to \infty} \paren {a_1^n + a_2^n + \cdots + a_r^n} = a$

Proof
From Sum of $r$ Powers is between Power of Maximum and $r$ times Power of Maximum:


 * $a^n \le a_1^n + a_2^n + \cdots + a_r^n \le r a^n$

and so:


 * $a \le \paren {a_1^n + a_2^n + \cdots + a_r^n}^{1 / n} \le r^{1/n} a$

From Limit of Integer to Reciprocal Power:
 * $n^{1 / n} \to 1$ as $n \to \infty$

Then for $n > r$:


 * $1 < r^{1 / n} < n^{1 / n}$

and so:
 * $r^{1 / n} = 1$ as $n \to \infty$

Thus as $n \to \infty$:


 * $a \le \paren {a_1^n + a_2^n + \cdots + a_r^n} \le a$

and the result follows by the Squeeze Theorem.