1. For each of the following sets, decide whether they are closed, bounded, compact (both) or neithe: (a) Q². (b) {1,2,3} x (0, 1].

Q1:- Let A, B, C, D, E be subsets of the Euclidean space R². Find their boundary, their interior, and their exterior. Conclude from here whether these sets are open, closed, or neither. a) A = {x € R² | d₂ (x,xo) ≤ 2}, where x₁ € R². b) B = Rx [a, b), where a, b E R, a < b. c) C = (a, b)² = (a, b) x (a, b), where a, b € R, a < b. d) D= (a) x [b, c), where a, b, c € R,b< c. e) E = {a} x {b, c), where a, b, c E R, b = c.

Show that a union of any finite points of bounded subsets of a metric space is bounded

Using the set definition, prove for any set X, Y and Z apply

Do the following task.a) State formally the Heine-Borel theorem in R^kb) Apply Heine-Borel theorem to show that if A is bounded, B isclosed such that B ⊆ A ⊆ R^k, then B is compact.c) Give an example of set A and B in R such that A is bounded,B is closed, B ⊆ A but A is not compact.

Evaluate using theorems on limits of 2 z²+1 6 z +1