Prove the following statements for n in N using: 1) The Principle of Mathematical Induction. 2) Computation. i) Sum from k=1 to n of k equals (n(n+1))/2. ii) Sum from k=1 to n of |... Prove the following statements for n in N using: 1) The Principle of Mathematical Induction. 2) Computation. i) Sum from k=1 to n of k equals (n(n+1))/2. ii) Sum from k=1 to n of |k| is less than or equal to 2√n. Given various properties related to real numbers. Solve the equations involving square roots and floor functions. Read more

Steps to Solve

1. Understanding the Tasks The problem consists of several exercises focusing on properties of real numbers, mathematical induction, inequalities, and sets. We need to address each exercise separately.

2. Exercise 1: Mathematical Induction

   • i) Prove that ( \sum_{k=1}^{n} k = \frac{n(n + 1)}{2} ) by induction.
     • Base case: For ( n = 1 ), ( LHS = 1 ) and ( RHS = \frac{1(1 + 1)}{2} = 1 ).
     • Inductive step: Assume true for ( n ) (i.e., it holds), then for ( n + 1 ), $$ \sum_{k=1}^{n+1} k = \sum_{k=1}^{n} k + (n + 1) = \frac{n(n + 1)}{2} + (n + 1) = \frac{(n + 1)(n + 2)}{2} $$
   • ii) Prove that ( \sum_{k=1}^{n} k^2 \leq 2\sqrt{n} ). Use inequalities and properties of squares.

3. Exercise 2: Properties of Real Numbers

   • i) Prove for any ( x, y \in \mathbb{R} ): ( |x| \leq a ) implies ( -a \leq x \leq a ).
   • ii) Find solutions to equations ( |x - 2| = 3 ).

4. Exercise 3: Floor Function

   • Determine ( \lfloor x \rfloor ) for ( x = [n] ), where ( n ) is an integer.
   • Prove properties involving ( x ) and floor functions.

5. Exercise 4: Analyzing Subsets of ( \mathbb{R} )

   • Identify which given sets ( A_1, A_2, \ldots, A_7 ) are intervals.
   • Calculate suprema and infima for the sets, justifying your reasoning.

6. Exercise 5: Bounded Sets

   • 1) Prove the set ( B ) is bounded above.
   • 2) Prove that ( \sup(B) = \sup(A) - \inf(A) ) using properties of suprema and infima.