= ｢n!/((n−i)!·i!)·1˄(n−i)·(1/n)˄i Σi=0,n｣
= ｢(1/i!)·n!/(n−i)!·(1/n˄i) Σi=0,n｣
= ｢(1/i!)·｢(n−j) Πj=0,i−1｣·(1/n˄i) Σi=0,n｣
= ｢(1/i!)·｢(n−j)/n Πj=0,i−1｣ Σi=0,n｣
= ｢(1/i!)·｢(1−j/n) Πj=0,i−1｣ Σi=0,n｣
< ｢(1/i!)·｢(1−j/(n+1)) Πj=0,i−1｣ Σi=0,n｣
< ｢(1/i!)·｢(1−j/(n+1)) Πj=0,i−1｣ Σi=0,n｣
+ (1/(n+1)!)·｢(1−j/(n+1)) Πj=0,(n+1)−1｣
= ｢(1/i!)·｢(1−j/(n+1)) Πj=0,i−1｣ Σi=0,n+1｣
= (1+1/n)˄(n+1)
｢(1+1/n)˄n｣ is increasing.
(1+1/n)˄n
= ｢(1/i!)·｢(1−j/n) Πj=0,i−1｣ Σi=0,n｣
< ｢(1/i!) Σi=0,n｣
｢(1+1/n)˄n｣ is bounded.
┄
A bounded monotone sequence is convergent.
┄
｢(1+1/n)˄n｣ is convergent.
