Opinions on infinitely close things?

[Radarithm]

What are your opinions on infinitely close objects / numbers?
I believe in them, because I can say 3.9 is close to four, but 3.999 is as well, and so is 3.99999999.
I've heard of LOTS of scientists who disagree with this, can anyone prove this wrong (or right)?

 

[CompuChip]

I would agree that 3.9 is close to four, and 3.999 is closer to it even, but no finite amount of 9's will make it equal to four.

I suggest reading https://www.physicsforums.com/showthread.php?t=507001 first, then we'll talk :)

 

[jbunniii]

The distance between two nonequal real numbers is a positive real number. Since there is no smallest positive real number, there is no number $x$ "infinitely close to" $y$ with $x \neq y$.

Proof that there is no smallest positive real number: you supply your candidate $x > 0$, I'll respond with $x/2$.

 

[economicsnerd]

It depends on how you define "number".