- #1
Artusartos
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In this link:
http://math.berkeley.edu/~scanez/courses/math104/fall11/homework/hw7-solns.pdf
In number 3, the text says "By the denseness of Q in R, there exists a sequence [itex](r_n)[/itex] of rationals converging to x."
I have several questions about this:
1) Why/how does denseness imply that there exists a sequence? Becuase Q is dense in R, we know that any two real numbers have a rational number between them. But how does that tell us that there is a "sequence" of rational numbers?
2) Do we have to assume that x is an irrational number?
3) Does this statement mean that for every irrational number, there is a sequence of rational number converging to it?
Thanks in advance...
http://math.berkeley.edu/~scanez/courses/math104/fall11/homework/hw7-solns.pdf
In number 3, the text says "By the denseness of Q in R, there exists a sequence [itex](r_n)[/itex] of rationals converging to x."
I have several questions about this:
1) Why/how does denseness imply that there exists a sequence? Becuase Q is dense in R, we know that any two real numbers have a rational number between them. But how does that tell us that there is a "sequence" of rational numbers?
2) Do we have to assume that x is an irrational number?
3) Does this statement mean that for every irrational number, there is a sequence of rational number converging to it?
Thanks in advance...