In Rudin's "Principles of Mathematical Analysis" he gives the strict definition for a subsequence as follows:(adsbygoogle = window.adsbygoogle || []).push({});

Given a subsequence {p}, consider a sequence {_{n}n}of positive integers, such that_{k}n<_{1}n<_{2}n<... Then the sequence {_{3}p} is called a_{ni}subsequenceof {p}. If {_{n}p} converges, its limit is called a_{ni}subsequential limitof {p}._{n}

Just underneath this definition, he says:

It is clear that {p} converges to_{n}pif and only if every subsequence of {p} converges to_{n}p.

From this definition alone, the answer is clear. Given any numerical sequence, basically if i start at anynterm and proceed infinitely (assuming that is what Rudin meant by "...") then I have a subsequence. But there are other definitions later in the book which recall this one and make it seem like a subsequence is any sort of segment of a sequence which can have a finite end and beginning, for instance, the following theorem which comes only one page after the previous definition:^{th}

The subsequential limits of a sequence {p} in a metric space_{n}Xform a closed subset ofX.

To me, this implies that there are (or at least, could be) multiple subsequential limits. However, the previous definition and the remark just after imply there is only one. Clearly I am misunderstanding something here. Does anyone have the insight I seem to be lacking? Thanks in advance.

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# What is the intuitive definition of a subsequence?

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