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$A = \{\sum_{k=1}^{\infty} \frac{a_k}{5^k}\}$ where $ a_k = [0,1,2,3,4]$ can the set $A$ contain an open interval? How?

jnyan
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2 Answers2

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Hint 1: Suppose the problem said $\sum_{k=1}^{\infty} \frac{a_k}{10^k},$ and the $a_k$ all belonged to $\{0,1,2,3,4,5,6,7,8,9\}.$ Could you solve it then?

Hint 2: For your problem, use base $5$ instead of base $10.$

Mitchell Spector
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  • I want elements of base 10, how would that be achieved? – jnyan Oct 31 '16 at 07:08
  • So every open interval in base 10, could be made in base 5, and that interval would still be open. It is like the whole real number line is written in base 5 form. Am I right? – jnyan Oct 31 '16 at 07:27
  • Real numbers are what they are. It doesn't matter how you write them--base 10, base 5, or something else. In the same way, there's no such thing as an open interval "in base 5" or "in base 10"; an open interval is just an open interval of real numbers, and you can write those real numbers in any convenient way. (The expression $\sum_{k=1}^{\infty} \frac{a_k}{10^k}$ is basically a decimal, and the expression $\sum_{k=1}^{\infty} \frac{a_k}{5^k}$ is the same thing but in base 5. So base 5 is convenient for thinking about your problem. The sums can be viewed as just ways of writing real numbers.) – Mitchell Spector Oct 31 '16 at 16:22
  • But you can't write 9 in base 5. It would be 14 in base 5. So when one looks at the real line, is it just a collection of things, which can be represented in different ways? I am sorry, all this time I have looked at real line in only base 10. – jnyan Oct 31 '16 at 16:25
  • Yes, the same point on the number line that you label as $"!9!"$ in base $10$ is labeled as $"!14!"$ in base $9.$ But those are just names; they represent the same number (essentially in different codes). It's like writing "nine" in English, "neuf" in French, or "nueve" in Spanish; they're just different ways of writing the same number. (In your example, though, you want to look at decimals, or the equivalent in base $5$ -- numbers between $0$ and $1.)$ – Mitchell Spector Oct 31 '16 at 16:30
  • Thanks a lot. One last thing, the question is equivalent to whether I can write all Real numbers between 0,1 if I were to use 0-9 digits and 10 in denominator. Correct? – jnyan Oct 31 '16 at 16:36
  • Yes, exactly -- those are just decimals, and your problem is essentially the same thing in that you're writing all the numbers between 0 and 1 as a "decimal" in base 5, using the digits 0 through 4. – Mitchell Spector Oct 31 '16 at 16:53
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Hint: An element of $A$ is a number (written in base $5$). Which numbers are achieved?

Stahl
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