Why is the supremum this?

Question

$M=\{y\in \mathbb R |y=3x+10:x\in(9,14)\}$

From what I've learned, this means $9<x<14$?

DEF: Supremum

A figure $u\in \mathbb R$ so

$a\leq u$ for all $a\in A$
for all $\epsilon>0$ there exists such $a\in A$ so $u-\epsilon<a$

How come 52 is said to be the supremum..? I would like to say that 49 is.

Sorry for the bad layout - I don't really know how to use LaTeX

$x\in (9,14) \Rightarrow 9<x<14 \Rightarrow 37<3x+10<52$, may be you can tell why you think 49 is supremum — Nick, Nov 09 '16 at 08:54
Because 52 - (a natural number - lets say 1) is 51. And 51 is not smaller than any a in M (biggest being 49... right?) — user1768788, Nov 09 '16 at 08:56

Noah Schweber · Answer 1 · 2016-11-09T08:59:07.193

3

Remember that $x$ doesn't have to be an integer! For example, take $x=13.9999$. Then $y=3x+10$ will be very very close to $52$ (specifically, $51.9997$).

edited Nov 09 '16 at 08:59

answered Nov 09 '16 at 08:57

Noah Schweber

245,398

Ah... This clears up a lot for me, thanks! – user1768788 Nov 09 '16 at 08:58
Thanks for identifying the source of OP's confusion. – MJD Nov 09 '16 at 12:53

score 0 · Answer 2 · answered Nov 09 '16 at 08:57

0

For $x \in (9,14)$ we have

$3*9+10 <3x+10<3*14+10$,

hence $37<3x+10<52$, therefor

$M=(37,52)$

answered Nov 09 '16 at 08:57

Fred

77,394

score 0 · Answer 3 · answered Nov 09 '16 at 08:58

0

First, notice that $M=(37,52)$
Supremum is defined to be the least upper bound.
For $M$, all numbers $\geq 52$ is an upper bound.
The least one is $52$, so supremum $=52$.

answered Nov 09 '16 at 08:58

Brian Cheung

1,977

Why is the supremum this?

3 Answers3