Can Natural Numbers m and n Satisfy x + 1/m < y - 1/n?

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No Effort - Member reminded that some effort must be shown in homework problems.
Homework Statement
prove x+1/m < y-1/n
Relevant Equations
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Let x, y ∈ R be such that x < y. Prove that there exist natural numbers m and n such that x +1/m < y −1/n?
 
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Because this is a homework-type of problem, you must show your work and then we can give hints and guidance. I will cheat and give you an initial hint. Consider y-x and see what that tells you about n and m.
 
Can you make an attempt at this?
 
FactChecker said:
Because this is a homework-type of problem, you must show your work and then we can give hints and guidance. I will cheat and give you an initial hint. Consider y-x and see what that tells you about n and m.
 
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i try but i do not know how to complete and i think my assumption is false
ass real.jpg
 
PeroK said:
Can you make an attempt at this?
ass real.jpg

i think my assumption is false i don ot know how i complete
 
Please type yoru attempt into a post, with some explanation of what you are doing.
 
For your proof to work, you would need to begin with ##\frac{1}{m}+\frac{1}{n}<0##, which is false for any ##m,n>0##.

If you get stuck, try proving it differently. Can you choose ##m## and ##n## such that ##\frac{1}{m}+\frac{1}{n}## is less than ##y-x##?
 
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by archimedean property right ? there is n,m which are integers s.t n>1/y and 1/m>-1/x then 1/n<y and 1/m<-x , add them we will get 1/m+1/n <y-x.
 
  • #10
mathstudent1 said:
by archimedean property right ? there is n,m which are integers s.t n>1/y and 1/m>-1/x then 1/n<y and 1/m<-x , add them we will get 1/m+1/n <y-x.
At some point, you'll need to learn to structure your arguments properly. That argument doesn't work as ##m, n## must be natural numbers. In particular, ##\frac 1 m## and ##\frac 1 n## cannot be negative.
 
  • #11
PeroK said:
At some point, you'll need to learn to structure your arguments properly. That argument doesn't work as ##m, n## must be natural numbers. In particular, ##\frac 1 m## and ##\frac 1 n## cannot be negative.
you are right , then how i prove it ?
 
  • #12
mathstudent1 said:
you are right , then how i prove it ?
My general advice if you are not sure how to prove something is to justify that it is true. In this case, you could start with, say, ##x = 0## and ##y = 1## and find suitable ##m, n##. Can you do that to start with?

Note that if you were more experienced, you might see a way to generalise that simple case to cover the case for any ##x## and ##y##.
 
  • #13
yes i can take m=3 and n=2
 
  • #14
mathstudent1 said:
yes i can take m=3 and n=2
If ##y - x \ge 1##, then those would do in general.

What about the case where ##y - x < 1##?
 
  • #15
PeroK said:
If ##y - x \ge 1##, then those would do in general.

What about the case where ##y - x < 1##?
they said in question x<y ,
PeroK said:
If ##y - x \ge 1##, then those would do in general.

What about the case where ##y - x < 1##?
also i can found m and n but i do not know who i sketch the proof
 
  • #16
mathstudent1 said:
they said in question x<y ,
Yes, but if ##x < y##, then ##y - x > 0##. That splits into two cases:

Case 1: ##y - x \ge 1##, where you have the solution ##m = \frac 1 3, n = \frac 1 2##.

Case 2: ##0 < y-x < 1##. Which looks harder, because ##y-x## is "small".
 
  • #17
Rewrite x+1/m < y-1/n as 1/m+1/n < y-x by subtracting x from both sides and adding 1/n to both sides.
So the problem can be restated as proving that you can find Natural numbers n,m such that 1/m+1/n < y-x.
Now, how can you make 1/m+1/n small enough? (Remember that x < y, so 0 < y-x.)
 
  • #18
mathstudent1 said:
also i can found m and n but i do not know who i sketch the proof
Let me give you a bit more help. When ##x =0## and ##y = 1##, here's what I think you probably did:

Note that ##\frac 1 2## is half way between ##0## and ##1##. So, let's take ##m, n \ge 2##. Although, it might have been better to take ##m = n = 3##.

For general ##x, y## you could generalise this:

Note that ##\frac {y- x}{2}## is half the distance between ##x## and ##y##. This is where you can apply the Archimedean property and take ##\frac 1 m = \frac 1 n < \frac{y-x}{2}##.
 
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  • #19
PeroK said:
Let me give you a bit more help. When ##x =0## and ##y = 1##, here's what I think you probably did:

Note that ##\frac 1 2## is half way between ##0## and ##1##. So, let's take ##m, n \le \frac 1 2##. Although, it might have been better to take ##m = n = \frac 1 3##.
. . .
@PeroK ,
Don't you mean ##\displaystyle \frac 1 m ## and ##\displaystyle \frac 1 n ## rather than ##m## and ##n## in this post ?
 
  • #20
FactChecker said:
Rewrite x+1/m < y-1/n as 1/m+1/n < y-x by subtracting x from both sides and adding 1/n to both sides.
So the problem can be restated as proving that you can find Natural numbers n,m such that 1/m+1/n < y-x.
Now, how can you make 1/m+1/n small enough? (Remember that x < y, so 0 < y-x.)

You can take n = m, so that you need \frac1{n} &lt; \frac{y - x}{2}.
 
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  • #21
SammyS said:
@PeroK ,
Don't you mean ##\displaystyle \frac 1 m ## and ##\displaystyle \frac 1 n ## rather than ##m## and ##n## in this post ?
Yes. Fixed, thanks.
 
  • #22
pasmith said:
You can take n = m, so that you need \frac1{n} &lt; \frac{y - x}{2}.
That is a nice simplification that allows one to get a formula for n (=m).
##n > \frac {2}{y-x}##, and ##m=n##.
 
  • #23
It is sufficient to require <br /> \frac1n + \frac1m \leq \frac 2{\min(n,m)} &lt; y - x where the first inequality is true for all positive n and m. This again leads to the problem of proving the existence of a positive N such that N &gt; \frac{2}{y - x}.
 
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