通过巧妙地构造辅助数列，应用致密性定理、柯西收敛准则来证明闭区间上连续函数的介值性定理。
We proved the intermediate value theorem for continuous function at closed interval by constructing auxiliary sequence ingeniously and applying compact theorem as well as Cauchy convergence criterion.
用数学分析中的区间套定理证明了闭区间上连续函数的四个定理。
Four theorems about continuous function on an closed interval are proved by a interval sequence theorem in mathematical analysis.
最后还证明了对于实数的子集a，如果定义在A上取值于A内的任一连续函数都有不动点，则A为实数的有限闭区间。
And finally we prove that let a be a subset of real Numbers, if every continuous function which is defined on a and taken value in a has fixed point, then a is a finite closed interval.
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