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 q not = p

Def. The statement that T is a subsequence of the sequence S means there is an increasing sequence, I, of positive integers such that T=SI

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

The question is: If the sequence S is bounded, then S has a subsequence T which has a limit

Explanation / Answer

I have written the answer on a paper and uploaded it as 4 pictures at the following links;

Part-1: http://i.imgur.com/78d9XE6.jpg

Part-2: http://i.imgur.com/idiCWhU.jpg

PArt-3: http://i.imgur.com/6znQtCy.jpg

Part-4: http://i.imgur.com/z7n8yLm.jpg

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