0.999 =1 is LUB or exactly the same

[Algorithm]

I want to ask that we say 0.999...=1 because the 1 is the least upper bound (supremum) of 0.999... or are they exactly same numbers.

I ask this because i am confused with finding (the infimum) the largest lower bound of 1. Do you think it should be 0.999... or is 0.999...=1.

Thanks for your comments...

 

[TD]

If you have a set S = \left\{ {0.9,0.99,0.999,0.9999,...} \right\} then 1 is indeed the supremum. But 0.999... = 1, at least if you mean a 0 with an infinite number of decimal 9's.

You may want to read this topic.

 

[phoenixthoth]

I want to ask that we say 0.999...=1 because the 1 is the least upper bound (supremum) of 0.999... or are they exactly same numbers.

I ask this because i am confused with finding (the infimum) the largest lower bound of 1. Do you think it should be 0.999... or is 0.999...=1.

Thanks for your comments...

The terms l.u.b. or g.l.b. are for sets, not individual numbers.

E.g., the l.u.b. of [0,1) is 1. The g.l.b. of [0,1) is 0.

The l.u.b. and g.l.b. of {1} is 1.

 

[Werg22]

1/3= 0.333...

Can be expressed by the serie

3/10 + 3/100 + 3/1000...

1/3 * 3 = 1

so

3*(3/10 + 3/100 + 3/1000...)

9/10 + 9/100 + 9/1000...

That is 0.999...

so 0.999... = 1

 

[HallsofIvy]

I can't believe this is here! I just responded to a new post in the "archives" about this same topic! 0.9999... is exactly equal to 1! Yes, 1 is also the supremum of the set {0.9, 0.99, 0.999, ...} but that is exactly the same as saying 0.999... = 1!