neter of a subset A mks) ii) A bounded subset A.of a metric space (X,d) (3 mks) b) Prove that the function, f(x)=x^2-3x is continuous at x x=Iunder the usual metric din't f'(x)-f'(y) 1-3 (+mks Q2 i) Show that the distance function d(x,y)=vert x_(1)-y_(1)vert +vert x_(2)-y_(2)vert for all x.y in R^2(x^2-3x) defines a metric in 9?^2 (4 mks) on X. ii) Let X=[0,1.2] with the standard metric don y_(i) Prove that f(x)=(1)/(4)x^2+(3)/(4) a contraction (4 mks) iii) Verify Banach's fixed theorem using the map in ii) (2 mks) (2) i) Explain what is meant by a compact subset A of metric space, (X, d) (4 mks) ii) Show that the set A=(0,2) is not compact in H_(1) (4 mks) iii) Determine whether the set of integers 2 is of first category or not (2 mks) f'(x)-f(y)

It seems like there are multiple parts to this question. Let's break it down and address each part one by one.### Part 1: Bounded Subset**Statement:**A subset of a metric space \( (X, d) \) is bounded if there exists a number such that for every point , the distance between and some point is less than .**Proof:**To prove that is bounded, we need to show that there exists an such that for every pair of points , the distance \( d(x, y) < M \).### Part 2: Continuity of the Function**Statement:**Prove that the function \( f(x) = x^2 - 3x \) is continuous at under the usual metric.**Proof:**To prove continuity at , we need to show that:1. \( f(1) \) is defined.2. The limit \( \lim_{x \to 1} f(x) \) exists.3. \( \lim_{x \to 1} f(x) = f(1) \).First, calculate \( f(1) \): Next, compute the limit as approaches 1: Since \( f(1) = -2 \) and \( \lim_{x \to 1} f(x) = -2 \), we have: Therefore, \( f(x) \) is continuous at .### Part 3: Distance Function Defining a Metric**Statement:**Show that the distance function \( d(x, y) = |x_1 - y_1| + |x_2 - y_2| \) for all defines a metric in .**Proof:**To show that is a metric, we need to verify the following properties for all :1. **Non-negativity:** \( d(x, y) \geq 0 \) and \( d(x, y) = 0 \) if and only if .2. **Symmetry:** \( d(x, y) = d(y, x) \).3. **Triangle inequality:** \( d(x, z) \leq d(x, y) + d(y, z) \).1. **Non-negativity:** - and , so \( d(x, y) = |x_1 - y_1| + |x_2 - y_2| \geq 0 \). - If , then and , so \( d(x, y) = 0 \).2. **Symmetry:** - \( d(x, y) = |x_1 - y_1| + |x_2 - y_2| = |y_1 - x_1| + |y_2 - x_2| = d(y, x) \).3. **Triangle inequality:** - For any , - Hence, \( d(x, z) \leq d(x, y) + d(y, z) \).Since satisfies non-negativity, symmetry,