推 yusd24 :用子序列做會比較容易一點 09/02 22:08
→ yusd24 :原PO這樣做也是沒錯的^^ 09/02 22:10
→ tianne :"子序列" 的方式是什麼意思呢?? 09/03 01:54
→ tianne :能不能大概說一下，因為不是很有概念，謝謝! 09/03 01:54
題目不難，但我証明的很爛(沒啥基礎，正在自修中)
希望各位不吝給我點意見和建議，謝謝!!
題目:
∞
Show that the sequence {Xn} in R-bar (Extended Real) is convergent if and
1
only if
lim sup (xn) = lim inf (xn)
n->∞ n->∞
我的想法:
(=>)
If this sequence is convergent, there is an x0 = lim {Xn}
n->∞
Since lim sup (xn) exists, lim |sup (Xn) - x0 | = 0.
n->∞ n->∞
Since lim inf (xn) exists, lim |inf (Xn) - x0 | = 0.
n->∞ n->∞
Therefore,
lim sup (Xn) = lim inf (Xn)
n->∞ n->∞
--------------------------------------------------------------------------
(<=)
If lim sup (Xn) ≠ lim inf (Xn)
n->∞ n->∞
Assume:
lim sup (Xn) = x0
n->∞
lim inf (Xn) = x1
n->∞
The limit of this sequence will be on [x0,x1], not convergent to a single
value.
Contradiction.
Therefore, this statement is false.
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 76.186.31.235