英语翻译海涅定理及其应用摘要：揭示了海涅定理的内涵,分别给出了不同函数极限的海涅定理,归纳总结了它的应用兵举出实例.关键

Maximum,progression,differential coefficient 哈哈哈,楼主,这样的翻译你要是用了,论文还能发表/通过毕业吗?
海涅定理及其应用 （Heine Theorem and its Application）
（作为标题,应该明白,实词首字母大写,虚词不用,这也是学术作品的一个小规范）
Abstract：This paper/dissertation explains the meaning of Heine Theorem,derives the different forms of the theorem for different functional limits,and summarizes the theorem's application with examples.
Keywords:Heine Theorem; series; function series; derivative
如果是要发表在刊物上论文,就用paper,毕业论文应当用dissertation.

Heine theorem and its applications
Abstract: Heine theorem reveals the meaning of the different functions ,gives Heine theorem with different functional limit , sums up the soldiers and gives examples of its application.
Keywords: Heine theorem; limit; series; functions; derivative

这是google翻译加上我个人修改后的结果：
Heine theorem and its applications
Abstract：This paper revealed the connotation of Heine theorem, made a list of Heine theorems in different function limits, summarized...

Heine theorem and its application
Abstract:This paper is intended to give a definition of Heine theorem in different functions limits and summarize its application through illustrating examples.
Key words:Heine theorem ;limit;progression;function sequence ;derivative

Heine Theorem and Its Application
Abstract: Reveled the connotation, showed the maximum of functions and summarized its application.
Key words: Heine Theorem, Maximum, progression, differential coefficient.

the Heine theorem and its application
abstract:reveal the connotation of the Heine theorem and different heine theorem with different function limit and summarize its application and give practical example
key words: Heine theorem,limit,series,function sequence,derivative