Proving Openness of a Set in n-Dimensional Space | Math Proof

[stauros]

Prove if the following set is open

\displaystyle{A=\begin{cases}<br /> \vec{x}=(x_1,...,x_n)\in\mathbb{R}^n:x_n&gt;0\\<br /> \end{cases}} .


I have written the following proof and please correct me if i am wrong

Let : \displaystyle{\vec{x}\in A}


Then we have : \displaystyle{\vec{x}=(x_1,...,x_n)} with \displaystyle{x_n&gt;0}

Choose \epsilon such that \displaystyle{0&lt;\epsilon&lt;x_n} and then \displaystyle{B(\vec{x},\epsilon)\subseteq A}


This happens because if \displaystyle{\vec{y}=(y_1,...,y_n)\in B(\vec{x},\epsilon)} then \displaystyle{||\vec{y}-\vec{x}||&lt;\epsilon}

and \displaystyle{y_i\in\left(x_i-\epsilon,x_i+\epsilon\right)}

Then we have \displaystyle{y_n\in\left(x_n-\epsilon,x_n+\epsilon\right)} and thus \displaystyle{y_n&gt;0}[/quote]

 

[HallsofIvy]

Prove if the following set is open

\displaystyle{A=\begin{cases}<br /> \vec{x}=(x_1,...,x_n)\in\mathbb{R}^n:x_n&gt;0\\<br /> \end{cases}} .


I have written the following proof and please correct me if i am wrong

Let : \displaystyle{\vec{x}\in A}


Then we have : \displaystyle{\vec{x}=(x_1,...,x_n)} with \displaystyle{x_n&gt;0}

Choose \epsilon such that \displaystyle{0&lt;\epsilon&lt;x_n} and then \displaystyle{B(\vec{x},\epsilon)\subseteq A}

That makes no sense because there is no one number labeled "x_n". What you mean to say is that \epsilon&lt; min(x_n).

This happens because if \displaystyle{\vec{y}=(y_1,...,y_n)\in B(\vec{x},\epsilon)} then \displaystyle{||\vec{y}-\vec{x}||&lt;\epsilon}

Can you prove this? That is, after all the whole point of the exercise! In particular, what is the definition of ||\vec{y}-\vec{x}||?

and \displaystyle{y_i\in\left(x_i-\epsilon,x_i+\epsilon\right)}

Then we have \displaystyle{y_n\in\left(x_n-\epsilon,x_n+\epsilon\right)} and thus \displaystyle{y_n&gt;0}

 

[stauros]

That makes no sense because there is no one number labeled "x_n". What you mean to say is that \epsilon&lt; min(x_n).



Can you prove this? That is, after all the whole point of the exercise! In particular, what is the definition of ||\vec{y}-\vec{x}||?

Yes you right,how about the inequality: \displaystyle{0&lt;\epsilon&lt;x_k, \forall 1\le k \le n}.

But i think the center point of the problem is that:

|x_{i}-y_{i}|\leq ||x_{i}-y_{i}||&lt;\epsilon using the Euclidian norm