高等数学定理的英语表达
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Theorem The sequence of functions {错误!未找到引用源。
}, defined on E, converges uniformly on E if and only if for every ɛ>0 there exists an integer N such that m ≥ N, n ≥ N, x 错误!未找到引用源。
E implies
错误!未找到引用源。
(1)
Proof Suppose {错误!未找到引用源。
} converges uniformly on E, and let 错误!未找到引用源。
be the limit function. Then there is an integer N such that n 错误!未找到引用源。
N, x错误!未找到引用源。
E implies
so that
if m ≥ N, n ≥ N, x错误!未找到引用源。
E.
Conversely, suppose the Cauchy condition holds. By Theorem 3.11, the sequence {错误!未找到引用源。
} converges, for every x, to a limit which we may call f(x). Thus the sequence {错误!未找到引用源。
} converge on E, to f. We have to prove that the convergence is uniform.
Let ɛ>0 be given, and choose N such that (1) holds. Fix n, and m错误!未找到引用源。
in (1). Since 错误!未找到引用源。
as m错误!未找到引用源。
, this gives
for every n错误!未找到引用源。
N and every x错误!未找到引用源。
E, which completes the proof.。
