How to prove a set is a bounded set?

[Cyn]

1. I have to show that
S1 = {x ∈ R2 : x1 ≥ 0,x2 ≥ 0,x1 + x2 = 2}
is a bounded set.2. So I have to show that sqrt(x1^2+x2^2)<M for all (x1,x2) in S1.3. I have said that M>0 and we have 0<=x1<=2 and 0<=x2<=2.
And x2 = 2-x1
We can fill in sqrt(x1^2 + (2-x1)^2) = sqrt (0^2 + (2-0)^2) = 2 < M = 3.
And we can fill in sqrt (x1^2 + (2-x1)^2) = sqrt (2^2 + (2-2)^2) = 2 < M = 3.
Every value between the 0 and the 2 that satisfy x1+x2 = 2 is smaller than this M. So the set is bounded.
Is this correct?

 

[mfb]

Every value between the 0 and the 2 that satisfy x1+x2 = 2 is smaller than this M.

Can you prove this?

The approach works, although I would make it easier: Just use 0<=x1<=2 and 0<=x2<=2 to find a maximal value the square root can get, and use that as limit. You don't have to find the smallest possible M.

 

[Ray Vickson]

1. I have to show that
S1 = {x ∈ R2 : x1 ≥ 0,x2 ≥ 0,x1 + x2 = 2}
is a bounded set.2. So I have to show that sqrt(x1^2+x2^2)<M for all (x1,x2) in S1.3. I have said that M>0 and we have 0<=x1<=2 and 0<=x2<=2.
And x2 = 2-x1
We can fill in sqrt(x1^2 + (2-x1)^2) = sqrt (0^2 + (2-0)^2) = 2 < M = 3.
And we can fill in sqrt (x1^2 + (2-x1)^2) = sqrt (2^2 + (2-2)^2) = 2 < M = 3.
Every value between the 0 and the 2 that satisfy x1+x2 = 2 is smaller than this M. So the set is bounded.
Is this correct?

Bounded usually means that $L_1 \leq x_1 \leq U_1$ and $l_2 \leq x_2 \leq U_2$ for some finite $L_1,L_2,U_1,U_2$ with $U_i \geq L_i$ for all $i$. Looking at $\sqrt{x_1^2 + x_2^2}$ is not necessary (but is harmless); one version of "bounded" is true if and only if the other is true as well. That is, the two versions are equivalent.

In your case you have $0 \leq x_i \leq 2$ for $i = 1,2$, and that is all you need. You can always enclose the square with vertices (0,0), (),2), (2,0), (2,2) inside a larger circle if you want to, and that will bound the norm $\sqrt{x_1^2 + x_2^2}$.