Let A denote a non-empty set. Explain what we mean by a sequence of elements in set A.
Thanks!
By "a sequence of elements in set A" we mean an ordered list of all the elements in A. That is, if we call the first element a_1, the second a_2, the third a_3 and so on, then the sequence of elements in set A is the sequence
{An}= a_1, a_2, a_3, a_4,....
for more on sequences see Sequence
Note that we can think of a sequence as a function with domain the natural numbers and range some subset of the real numbers. thinking about it that way, if we can find some function of n, where n represents the elements of the natural numbers, then An (that is A sub n), will be the formula for this sequence. plugging in n=1 into the function we get a_1, plugging in n=2 into the function we get a_2 and so on.
I need to warn you. Our Analysis book, accepts that set N (natural numbers) are {1,2,3...}.
In fact, the standard notion of natural number includes zero as well.
This is why I used Z^+.
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And when you said "subset" you should have said "non-trivial subset". Again our book does never says that.