Metric Space, Show that it's open

[szklany]

Let (X,d) be a metric space, and x is an element in X. Show that \{y \in X|d(y,x)>r\} is open for all r in Reals.

I really need some help with this one, I have almost no idea on how I am meant to solve this.

The only thing i know is that I have to use the Openness definition, that states something like \forall x_0 \in U \exists r>0| B_r \in U, where in U is a subelement of the metric space (X,d).

But i don't know how to get started.

 

[HallsofIvy]

Let (X,d) be a metric space, and x is an element in X. Show that \{y \in X|d(y,x)>r\} is open for all r in Reals.

I really need some help with this one, I have almost no idea on how I am meant to solve this.

The only thing i know is that I have to use the Openness definition, that states something like \forall x_0 \in U \exists r>0| B_r \in U, where in U is a subelement of the metric space (X,d).

But i don't know how to get started.

Let a be a point in {y| d(x,y)> r}. Then d(x,a)> r. Construct the neighborhood about a with radius (d(x,a)- r)/2. If b is any point in that neighborhood, use the triangle inequality to show that d(x, b)> r also.