求证：一群有限个正数的n次方之和的开n次方等于这群数中的最大者.

这个表述不太对,应该是n趋于无穷时的极限等于最大数,即:
lim{n → ∞} (a1^n+a2^n+...+am^n)^(1/n) = max{a1,a2,...,am}.
不妨设a1 = max{a1,a2,...,am}.
则a1 ≥ ak > 0,对k = 1,2,...,m成立.
于是a1^n ≤ a1^n+a2^n+...+am^n ≤ m·a1^n.
a1 ≤ (a1^n+a2^n+...+am^n)^(1/n) ≤ m^(1/n)·a1.
当n → ∞,有m^(1/n) → 1.
于是a1 ≤ lim{n → ∞} (a1^n+a2^n+...+am^n)^(1/n) ≤ a1.
即lim{n → ∞} (a1^n+a2^n+...+am^n)^(1/n) = a1 = max{a1,a2,...,am}.