數列有界不一定收斂.
例如 a_n = (-1)^n, 這個數列有界, 不收斂.
另一個例子: a_n = sin(n).
事實上有界而不收斂的多得是, 例子隨便就可舉一籮筐.
An= sin^2(n/2n+1)+cos^2(n/2n+1)
不管裡面的 n/2n+1 究竟是什麼東西, sin 與 cos 內的相同,
則 An ≡ 1, 因此當然是收斂的.
事實上, 如果裡面是 n/(2n+1), 那麼它收斂至 1/2.
因為 sin, cos 是連續函數, 因此,
lim sin(n/(2n+1)) = sin(1/2)
lim cos(n/(2n+1)) = cos(1/2)
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