Re: [高微] 證明數列收斂
※ 引述《IminXD (Encore LaLa)》之銘言：
: 題目:Let Sn be a bounded sequence of real numbers. Assume 2Sn≦S_n-1 + S_n+1
: Show that lim ( S_n+1 - Sn ) = 0
: n->∞
Proof. Let a_n ＝ s_(n+1) - s_n, it is easy to see that {a_n} converges since
it is increasing and bounded. The limit of {a_n} is denoted by a. We
will show that the limit a is zero as follows.
(1) If a ＞ 0, then there exists a positive integer n_0 such that as n ≧ n_0,
we have
a/2 ＜ a_n ＝ s_(n+1) - s_n.
It follows that s_(n_0) ＋ ka ＜ s_(n_0 ＋ 2k) which contradicts to the
boundedness of {s_n}.
(2) Similarly for a ＜ 0.
Hence, from above sayings we have proved the limit is zero. □
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