∵ sin²1+sin²2+……+sin²n > sin²1
sin²1+sin²2+……+sin²n < 1+1+.+1 = n
∴ (sin²1)^(1/n) < (sin²1+sin²2+……+sin²n)^(1/n) < n^(1/n)
设 un = (sin²1+sin²2+……+sin²n)^(1/n),对un取对数,则有：
2ln(sin1)/n < ln(un) < ln(n)/n
(n→∞)lim[2ln(sin1)/n] < (n→∞)lim[ln(un)] < (n→∞)lim[ln(n)/n]
∵ (n→∞)lim[2ln(sin1)/n] = 0； (n→∞)lim[ln(n)/n] =0
∴ 根据夹逼定理有：
(n→∞)lim[ln(un)] = 0 ==> (n→∞) lim(un) = e^0 =1；
结论：
(n→∞) lim[(sin²1+sin²2+……+sin²n)^(1/n)] =1