Let \[ A_1\ge A_2\ge\cdots\ge A_k\ge0. \] Evaluate \[ \lim_{n\to\infty}\left(A_1^n+A_2^n+\cdots+A_k^n\right)^{1/n}. \]
Solution
1. For every positive integer \(n\),\[ A_1 \le \left(A_1^n+A_2^n+\cdots+A_k^n\right)^{1/n} \le k^{1/n}A_1. \]
Since all \(A_j\) are nonnegative, the sum contains the term \(A_1^n\), so \[ A_1^n \le A_1^n+A_2^n+\cdots+A_k^n. \] Also \(A_j\le A_1\) for every \(j\), hence \[ A_1^n+A_2^n+\cdots+A_k^n \le kA_1^n. \] Taking nonnegative \(n\)th roots gives the displayed inequalities.
2. The required limit is\[ \boxed{A_1}. \]
Since \[ \lim_{n\to\infty}k^{1/n}=1, \] step 1 and the squeeze theorem give \[ \lim_{n\to\infty} \left(A_1^n+A_2^n+\cdots+A_k^n\right)^{1/n} = A_1. \] This also covers \(A_1=0\), in which case every \(A_j\) is \(0\).
3. Q.E.D.
Step 2 gives the requested value.