Sequence that has all rational numbers

In summary, the conversation is about constructing a sequence that contains all rational numbers in order to prove that every real number is its limit point. The original question is about constructing a sequence that has all real numbers as its limit point. Suggestions are given to try putting the rational numbers in an array and finding a path that goes through all of them, or to look for ideas in a resource provided.
  • #1
ask_questions
4
0

Homework Statement



Construct a sequence that has all rational numbers in it

Homework Equations



None.

The Attempt at a Solution



Here are my thoughts, though I have no solutions yet.

If I construct a sequence Sn= n*sin(n)-1/n, will it work?

Thanks guys!
 
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  • #2
The sine term will often give irrational numbers, so that won't work. Try putting the rationals in an array and finding a path that goes through all of them.
 
  • #3
thanks! can you elaborate a little bit? I'm trying to self-study real analysis, and I'm not really familiar with what you just mentioned...
how would i put them in an array and find a "path"?

thanks!
 
  • #4
ask_questions said:

Homework Statement



Construct a sequence that has all rational numbers in it

Homework Equations



None.

The Attempt at a Solution



Here are my thoughts, though I have no solutions yet.

If I construct a sequence Sn= n*sin(n)-1/n, will it work?

Thanks guys!
Just drop the sine term.

OR

Do you really want a sequence with all of the rational numbers in it. -- maybe just all of the positive rationals?

Better yet: Please type the problem word fro word as it was presented to you.
 
  • #5
Hi guys:

Thank you so much! Here's the problem as it was typed on the book:

Construct a sequence such that every real number is its limit point.

I know this is different from the question I typed above, but my reasoning is that if i can have a sequence that contains all rational numbers, then I can prove that every real number is its limit point. Does that make sense? How should I solve the original question if this does not?

Thank you!
 

What is a sequence that has all rational numbers?

A sequence that has all rational numbers is a list of numbers that includes every possible rational number. This means that the sequence contains fractions with integer numerator and denominator, as well as repeating decimals.

What is the pattern of a sequence that has all rational numbers?

The pattern of a sequence that has all rational numbers is that it follows the order of the rational numbers from least to greatest. This means that the numbers in the sequence will increase or decrease in value as you move through the list.

How is a sequence that has all rational numbers different from a sequence that has all whole numbers?

A sequence that has all rational numbers includes fractions and decimals, while a sequence that has all whole numbers only includes integers. This means that a sequence with all rational numbers has a larger range of values than a sequence with all whole numbers.

Can a sequence that has all rational numbers have a repeating pattern?

Yes, a sequence that has all rational numbers can have a repeating pattern. This can occur when the sequence contains repeating decimals, such as 0.333... or 0.666... The pattern would continue to repeat infinitely.

Is there a real-life application for a sequence that has all rational numbers?

Yes, a sequence that has all rational numbers can be used in various mathematical and scientific calculations. For example, in physics, rational numbers are used to represent measurements and quantities, making a sequence with all rational numbers useful in solving real-world problems.

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