Prove the Positive Orthant is Open (Proof Verification) I'm practicing my proof-writing and was hoping you could let me know if this proof looks good. I would like to know if the proof is incorrect, if there are parts that are overly wordy/complicated, or if I'm missing some element of a proof that is helpful to see, if not strictly necessary.
The Prompt:
Prove that the positive orthant R+m is an open subset of Rm.
My Proof:
Let x be an arbitrary element of R+m. Let ε = inf xi for all i = 1,...,m. Now define open ball Bε(x) = {y ∈ Rm | ||x-y|| < ε}. The closest point to outside of R+m will be in a straight line from x in the dimension of x which is closest to 0 (call it dimension k, so xk=inf xi). Thus, set all values of yi = xi except for when i=k. Then:
||x-y|| = sqrt((xk2-yk2)+0+0...) = xk - yk < ε
xk - yk < inf xi
xk - inf xi < yk
0 < yk
So yk > 0 and since yi = xi ∈ R+ for all other i, y ∈ R+m. Since any element of R+m has an open ball in R+m surrounding it, R+m is open.
 A: It is a good start, but then it starts getting a bit wordy / overcomplicated. Here are a few points that could make things smoother.


*

*You use $x_i$ without defining them. You may want to write "let $x = (x_1,\dots,x_m) \in \mathbb{R}_+^m$".

*Then, writing
$$
\epsilon = \inf_{i=1,...,m} x_i > 0
$$
would be better notation.

*You do not define the norm $\| \cdot \|$. Maybe it is given before, but we do not know this. Apparently, it is the Euclidean norm.

*The main issue is this mysterious sentence "The closest point to outside of $\mathbb{R}_+^m$ will be in a straight line from x in the dimension of x which is closest to $0$ (call it dimension k, so $x_k=\inf x_i$)" It is not clear what you mean here, and this relies a lot on two-dimensional intuition. What you really want to show is that the ball $B_{\epsilon}(x)$ is indeed included in the positive octant. This means, take
$$
y = (y_1,\dots,y_n) \in B_{\epsilon}(x),
$$
and show that $y \in \mathbb{R}_+^m$, i.e. that $y_i > 0$ for each $i$. Can you get this, using triangle inequality and your definition of $\epsilon$?

