Prove the Positive Orthant is Open (Proof Verification)

Question

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 R^m.

My Proof:
Let x be an arbitrary element of R₊^m. Let ε = inf x_i for all i = 1,…,m. Now define open ball B_ε(x) = {y ∈ R^m | ||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 x_k=inf x_i). Thus, set all values of y_i = x_i except for when i=k. Then:

||x–y|| = sqrt((x_k²-y_k²)+0+0…) = x_k – y_k < ε

x_k – y_k < inf x_i

x_k – inf x_i < y_k

0 < y_k

So y_k > 0 and since y_i = x_i ∈ 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.

Answer 1

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 $\epsilon$?