Basic topology (differing metric spaces in R^2)

[mjkato]

Got it, thank you

 

[Mark44]

Homework Statement

Let d1(x,y) = |x1 − y1| + |x2 − y2| and d2(x,y) = ((x1 − y1)^2+(x2 − y2)^2)^(1/2) be metric spaces on R^2.

d1 and d2 aren't metric spaces - they are metrics.

(R², d₁) is a metric space, and so is (R², d₁). In other words, a metric space consists of a set (such as R²) together with a metric (or measure).

Prove that if a point (x1,x2) is a limit point of a subset under one metric, it is a limit point in the other.

Homework Equations

The Attempt at a Solution

I'm mostly lost here- I know a limit point contains atleast one point from the set in every neighborhood, but can't figure out how to translate that into a proof of this.

How do you describe a neighborhood?