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Since every countably infinite set is the range of a bijective function defined on $N=\{1,2,3,\dots\}$, we may regard every countable set as the range of a sequence of distinct terms. Speaking more loosely, we may say that the elements of any countable set can be "arranged in a sequence."

Does the opposite statement make sense ? i.e. if we can arrange the elements of a set as the elements of a sequence of distinct elements (for example, if that set is an infinite subset of a countably infinite set, say $X$, then using the above argument for $X$) then that set is countably infinite. I am confused.

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How do you define "arrange in a sequence?" Most rigorous definitions would be as the $1-1$ range of a function on $\mathbb N$. –  Thomas Andrews Aug 14 '13 at 0:28

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That's exactly what "arranging in a sequence" is. You essentially create a bijective map from a $\mathbb{N}$ to a set $A$. To see this, lets say we arrange the elements of $A$ into the sequence $a_1, a_2, a_3, \dotsc$ with each $a_i$ unique. Then by using the indices $1, 2, 3, \dotsc$, you've implicitly defined the map $i \mapsto a_i$. To make it explicit, just define the function $a: \mathbb{N} \to A$ by $a(i) = a_i$. It is easy to see it is bijective since $a_i$ are unique.

So yes you can go the other way. If you can arrange the elements into a sequence of distinct terms, then the set is countable.

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