作者iddee (好痛哥哥)
看板Math
標題Re: [分析] 初微(12)
時間Sat Jul 16 01:57:57 2005
※ 引述《PttFund (批踢踢基金)》之銘言：
: Let {a_n} and {x_n} be two real sequences. If a_n > 0 for all n,
: 1
: lim ----------------- = 0,
: n→∞ a_1 + ... + a_n
: and x_n → x as n → ∞. Show that
: a_1 x_1 + ... + a_n x_n
: lim ------------------------- = x.
: n→∞ a_1 + ... + a_n
By assumption => for any ε>0, there is an integer N>0
such that n≧N => |x_n - x|<ε/2
a_1 x_1 + ... + a_n x_n M n a_i(x_i - x)
|------------------------- - x| ≦ --------------- + Σ |---------------|
a_1 + ... + a_n a_1 + ... + a_n i=N a_1 + ... + a_n
where M = |Σ a_i(x_i - x)|, (i from 1 to N-1)
< M/(a_1 + ... + a_n) + ε/2
for same ε, we can choose an integer N'≧N
such that n≧N' => |1/(a_1 + ... + a_n)|<ε/2M
then,
< ε/2 + ε/2 = ε whenever n≧N'
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