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※ 引述《sunfin (遠方)》之銘言：
: 1.年級：高一資優數學
: 2.科目：高中數學
: 3.章節：龍騰 第一冊第一章 集合問題
: 4.題目：
∞ ∞ ∞ ∞ ∞
Let {A_n} be a sequence of sets such that ∪ ∩ A_n = ∩ ∪ A_n.
n=1 k=1 n=k k=1 n=k
Which of the following sequenses has this property?
(a) A_n = { x屬於R | -1/n < x <= 1+(-1)^n }
(b) A_n = { x屬於R | -(1+n)/n < x < (1+n)/n }
(c) A_n = { x屬於R | -(1+n)/n <= x <= (1+n)/n }
[Sol]:
(a)
∞ ∞ ∞
A_1 = (-1 , 0] ∩ A_n = [0,0], ∪ ∩ A_n =[0,0]
A_2 = (-1/2, 2] n=k k=1 n=k
A_3 = (-1/3, 0]
A_4 = (-1/4, 2] ∞ ∞ ∞
A_5 = (-1/5, 0] ∪ A_n = (-1/k, 2], ∩ ∪ A_n =[0,2]
A_6 = (-1/6, 2] n=k k=1 n=k
This sequence has no such property.
(b)
∞ ∞ ∞
B_1 = (-2 , 2 ) ∩ B_n = [-1,1], ∪ ∩ B_n =[-1,1]
B_2 = (-3/2, 3/2) n=k k=1 n=k
B_3 = (-4/3, 4/3)
B_4 = (-5/4, 5/4) ∞ k+1 k+1 ∞ ∞
B_5 = (-6/5, 6/5) ∪ B_n = (---,---), ∩ ∪ B_n =[-1,1]
B_6 = (-7/6, 7/6) n=k -k k k=1 n=k
This sequence has this property.
(c)
∞ ∞ ∞
C_1 = [-2 , 2 ] ∩ C_n = [-1,1], ∪ ∩ C_n =[-1,1]
C_2 = [-3/2, 3/2] n=k k=1 n=k
C_3 = [-4/3, 4/3]
C_4 = [-5/4, 5/4] ∞ k+1 k+1 ∞ ∞
C_5 = [-6/5, 6/5] ∪ C_n = [---,---], ∩ ∪ C_n =[-1,1]
C_6 = [-7/6, 7/6] n=k -k k k=1 n=k
This sequence has this property.
這的確是 limsup 與 liminf
但只要對集合有基本的認識，老師都應該能看得懂並作出這題的答案
要讓學生知道這題目在幹嘛很簡單
就像我上面作的事情，一項一項帶進去給學生看，一項一項推導就好了
「看不懂數列在幹什麼，就帶個幾項進去看看」
這件事是一定要教給學生的
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