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.

$\blacksquare$

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $1$: Review of some real analysis: Exercise $1.5: 11$