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December 22nd, 2016, 05:12 PM  #1 
Member Joined: Oct 2016 From: New York Posts: 49 Thanks: 13 Math Focus: Analysis and Differential Geometry  If A is a set.. sequences of elements in A
What I know: If A is a set and $\displaystyle {x_n}$ is a sequence of elements of A then it's not necessarily true that the terms $\displaystyle x_1, x_2, x_3,...$ are distinct. What I'm confused on: The proof I am reading has a statement, "Arrange the elements of $\displaystyle x$ of A in a sequence $\displaystyle {x_n}$ of distinct elements." So I get that what they are basically saying is that we are putting the elements of A into sequence, in other words, loosely speaking we are giving each of them a number. My question is, are we assuming that all the elements of A are distinct? Do we assume that the phrase, "Let A be a set" means that they are distinct elements? If they are not distinct, it doesn't matter right? We still just make a sequence of all of the terms but we would treat the repeated elements as if they are distinct by giving them their own number. (By number I mean their own $x_n$ for some n). Sorry if this is an "obvious question" but I coudn't find a clear answer in any of the books I have. I am attaching a picture of the proof I am reading. Last edited by ProofOfALifetime; December 22nd, 2016 at 05:15 PM. 
December 22nd, 2016, 05:46 PM  #2 
Member Joined: Oct 2016 From: New York Posts: 49 Thanks: 13 Math Focus: Analysis and Differential Geometry 
Okay, so after I posted this I found the answer. We assume that they are distinct because $\displaystyle {2,3, 2, 3, 4, 5, ...}$would not be a valid set. Sorry everyone.

February 4th, 2017, 05:40 PM  #3 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,484 Thanks: 628 
The sentence "Arrange the elements x of A in a sequence xn of distinct elements" doesn't say that a general sequence has to have distinct elements, just that we can write all the elements of x in such a sequence. (This is true only for countable sets.) 

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