Question

Prove that d defined by d(x,y)=1+∣x−y∣∣x−y∣​,x,y∈R, is a metric on R. (II) (a) Construct a set with exactly two limit points. Explain your answer in detail

Answer 1

To prove that d defined by d(x, y) = 1 + |x - y| / |x - y|, x, y ∈ R, is a metric on R, we need to show that it satisfies the properties of a metric. To construct a set with exactly two limit points, we can take the set A = {0, 1} as an example.

Step-by-step explanation:

To prove that d defined by d(x, y) = 1 + |x - y| / |x - y|, x, y ∈ R, is a metric on R, we need to show that it satisfies the properties of a metric:

1. Non-negativity: d(x, y) ≥ 0 for all x, y ∈ R.
2. Identity: d(x, x) = 0 for all x ∈ R.
3. Symmetry: d(x, y) = d(y, x) for all x, y ∈ R.
4. Triangle inequality: d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ R.

To construct a set with exactly two limit points, we can take the set A = {0, 1} as an example. The limit points of A are 0 and 1; any neighborhood of 0 or 1 will contain infinitely many points of A.