Re: [分析] 初微(60)
※ 引述《plover (>//////<)》之銘言：
: If Σa_n converges with a_n > 0 for all n, and {a_n} is a
: decreasing sequence, show that n a_n → 0 as n → +∞.
By assumption => for any ε>0, there exists an integer N>0 s.t. n ≧ N
we have a_(n+1) + ... + a_(n+n) < ε/2
{a_n} is a decreasing seq. => n[a_(2n)] < a_(n+1) + ... + a_(n+n) < ε/2
Hence 2n[a_(2n)] < ε whenever n ≧ N
Similarly, a_(n+1) + ... + a_(n+n+1) < ε/2 whenever n ≧ N' for some N'
then (n+1)a_(2n+1) < a_(n+1) + ... + a_(n+n+1) < ε/2
so (2n+1)a_(2n+1) < (2n+2)a_(2n+1) < ε
Hence n(a_n) < ε whenever n ≧ max{N,N'}
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