→ wheniam64 :感謝大大!! 08/17 23:55
※ 引述《wheniam64 (嘿)》之銘言：
: 1. Suppose that an →L. Show that if an 小於等於 M for all n,
: then L 小於等於 M .
Assume the contray, that is, L > M. Since a_n -> L as n -> ∞, given
ε = (L-M)/2 > 0, there is an integer N such that |a_n - L| < (L-M)/2.
Then a_n > L - (L-M)/2 > M, contradicts the fact that M is an upper bound
of {a_n}.
: 2. Let f be a function continuous everywhere and let r be a real number.
: Define a sequence as follows:
: a1=r , a2=f(r) , a3=f(f(r)) , .........
: Prove that if an→L, then L is a fixed point of f:f(L)=L
Since f is a continuous function on |R, for any sequence a_n -> a (real)
we have f(a_n) -> f(a). From the given problem, we know that a1 = r and
a_(n+1) = f(a_n). Letting n -> ∞, lim(n->∞) a_(n+1) = lim(n->∞) f(a_n)
or L = f(L) by the continuity of f at real number L.
: 這二題習題和同學討論許久不得其解
: 請版上高手解答
: 感謝!
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