推 cxcxvv :謝謝~ 09/18 21:48
※ 引述《cxcxvv (delta)》之銘言：
: Show that
: if X_n - X_(n-2) → 0 (n→∞)
: then X_n/n → 0
X ＝ (X － X ) ＋ (X － X ) ＋ ... ＋ (X － X ) ＋ X
n n n-2 n-2 n-4 k+2 k k
where k = 1 if n is odd and k = 2 if n is even. Since
X － X → 0 as n→∞, for any ε ＞ 0, we can find N
n n-2
ε
such that │X － X │ ＜ ── for all n ≧ N. Next, we can find
n n-2 2
│X_k＋( X_(k+2)－X_k )＋...＋( X_N－X_(n-2) )│
natural number M such that │──────────────────────│
│ n │
ε
＜ ── for all n ≧ M ＞ N.
2
│ X_n │ │ X_k ＋ ( X_(k+2) － X_k ) ＋ ... ＋ ( X_N －X_(N-2) )│
So │───│ ＜ │───────────────────────────│
│ n │ │ n │
[n － N] ε ε
＋ ───── ε ＜ ── ＋ ── ＝ ε for all n ≧ M.
2n 2 2
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