"Prove that a function d: A × A → R is precisely a metric on the set A if and only if: 1. For every x, y Є A, d(x, y) = 0 if and only if x = y. and 2. For all x, y, z Є A‚d(x, y) ≤ d(z, x) + d(y, z)."

A. Let H be the set of all points (x, y) in ℝ2 such that x2 + 3y2 = 12. Show that H is a closed subset of ℝ2 (considered with the Euclidean metric). Is H bounded?

Let (X,d) be a metric space and   d*(x,y) = d(x,y)/1+d(x,y) Prove that the family of open sets with respect to the metric d is same as the family of open sets with respect to the metric d*.

Consider a set A and a function d: A × A → R that satisfies:• d(x, y) = 0 ⇔ x = y;• d(x, y) = d(y, x);• d(x, y) ≤ d(x, z) + d(z, y).Prove that (A, d) is a metric space, i.e. show that d(x, y) ≥ 0 forall x, y ∈ A.

2.) Let (S, d) be a metric space and suppose that ρ : S × S → R is defined byρ(x, y) = d(x, y)1 + d(x, y)for all points x, y ∈ S. Prove that (S, ρ) is a metric space, that it is bounded and thatρ(x, y) ≤ d(x, y) for all x, y ∈ S.

Let (X,d) be a metric space and E ⊆ X. Prove that if E is compact, then E is bounded.

b)Let E be a non-empty subnset of a metric space(X,d),define the distance of x fromE by: ρE(x)=inf∈Ed(x,z).

On a previous homework, you proved a Bolzano-Weirstrass theorem for ℝ3 with the metric d((x1, x2, x3), (y1, y2, y3) = |x1 - y1| + |x2 - y2| + |x3 - y3|. Conclude that, with this metric, a subset of ℝ3 is sequentially compact if and only if it is closed and bounded.

A. Let H be the set of all points (x, y) in ℝ2 such that x2 + xy + 3y2 = 3. Show that H is a closed subset of ℝ2 (considered with the Euclidean metric). Is H bounded?A. Let H be the set of all points (x, y) in ℝ2 such that x2 + xy + 3y2 = 3. Show that H is a closed subset of ℝ2 (considered with the Euclidean metric). Is H bounded?

Is the set S = [0,1] with the discrete metric d separable? Explain.