Def. The statement that S is a sequence means that S is a function with domain s

Question

Def. The statement that S is a sequence means that S is a function with domain some initial segment of the positive integers. (That is: the domain of S is either the set of positive integers or the domain of f is the set {1,2,3,...,n} for some positive integer n.)

Def.     The statement that the number, p, is a limit point of the number set, A, means that if (a,b) is an open interval containing p, (that is: a<p<b) then there is a number q in A such that a<q<b, and    (undefined) Def. The statement that T is a subsequence of the sequence S means there is an increasing sequence, I, of positive integers such that   

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Def. The image of a sequence, denoted Im(S) for the image of sequence S, is the set of all numbers in the sequence

Def. if D is a subset of R and G is a collection of open intervals such that if x is in D then x is in g for some gin G, then G is an open cover of D, A subset of G also covers d is called a subcover.

Def. A subset D of R is compact if every open cover has a finite subcover

The question is:

The sequence S has limit p if and only if each subsequence T has a limit p

Explanation / Answer

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Question-1: http://i.imgur.com/DLmUEI4.jpg

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