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

Answer 1

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

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

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

$M=(37,52)$

Answer 2

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, $51.9997$).

Answer 3

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 $=52$.