m above, prove that for every real number c>0, there is a 1/N < c.

Q: Q3 Prove that In Coth x 1

A:

Q: Prove: | a | - | b | &lt;= | a + b |

A:

Q: Find all equilibrium or constant solutions (if any) to the ODE below y" = ecost/ – /2ty' + y? – 1…

A: In this question, constant solutions are to be determined. Constant Solutions When the derivative of…

Q: 2. Show that for any real number c, there is exactly one integer in the interval (c,c + 1].

A: As per guidelines, I can do only first question.

Q: -. For any integer not an integer, then [x]+[m- x] = m-1. m and any real number x, if x is

A: Let m be any integer and x be any real number that is not an integer.

Q: Let 0 < a < 1 be a real number. Define a₁ = 1+a, Prove that Vn € N, 1 ≤n, an+1 = 1<an< 1 - an + a, -…

A:

Q: Let a,b and x be real numbers with a0. Show that there exists a rational number t such that a <

A: We know that the real numbers are densely ordered. This means that between any two distinct real…

Q: (∀x)(x < 1 → x2 < x). (For all positive real numbers less than 1, squaring that number actually…

A:

Q: Let n be an odd integer, a > 0, and b < 0. Prove that x" + ax + b= 0 has a unique root. Hint: If…

A: Let n be an odd integer, a &gt; 0, and b &lt; 0. Prove that xn + ax + b = 0 has a unique root

Q: -. For any integer m and any real number x, if x is not an integer, then [x]+[m- x] = m-1.

A:

Q: Find a given that ||a i+ (a – 1)j + (a + 1)k|| = 7.

A: Given:

Q: Prove that for all positive real a we have 22 +5 20x4 2 4/5. 24 x2 +5

A:

Q: () is even. 3. Prove that for all positive integers n, the number

A:

Q: Need

A: Given that, a can be written as base 10 x= a0+10a1+ ......+10nar We know that, If a divides b…

Q: 11. Prove that every positive integer n can be expressed in the form n=4k or n=4k+1 or n=4k+2 or…

A:

Q: .18 Let x, a and 8 be real numbers. Show that the statements |x-al < 8 and a -8 < x < a +8 are…

A:

Q: Let x be a real number. Prove the following statement: for each e > 0, there exists a rational…

A: Given  that x be a real number. And we have to prove the following statement: for each ε&gt;0, there…

Q: Prove that the roots of X2+(1-K)x+k-3=0 are real for all values of k?

A:

Q: Consider the following proposed proof, which claims to show that every nonnegative integer power of…

A: Note: we know that in strong induction principle, we assume that the particular statment holds at…

Q: Prove or disprove: There exists a positive real number x such that x² <√√x-

A: Given statement isThere exist a positive real number x such that  x2&lt;x.This is an existential…

Q: 1. Let a be a positive real number and n be a natural number. Define S = {x €R: x >0 and x" a?…

A:

Q: Let a1, a2, ...,a, be real numbers that are all contained in the interval [-7/2, 7/2]. Use the…

A:

Q: =   x ∈ R  1 ≤ x ≤ 1 +  1 i   =   1, 1 +

A: If x varies from -1 to 1 then x+1 varies from 0 to 2.

Q: If A = 0 and Ak = 0, for some posit ive integer k , then A must be singular Select one: O True O…

A: The solution is :

Q: Which of the following statement is always false? (a) For all integer x, [x] = [-x] (b) For all odd…

A:

Q: Suppose that x is a negative real number and that x < 1/x. Prove that x < 1.

A: Step 1: Step 2: Step 3: Step 4:

Q: Which of the following statement is always false? (a) For all integer x, [x] = [−x] (b) For all odd…

A: We have to check which of the given option is always false. a Given that  for all integers x,  x=−x.…

Q: Let n be a real number and let E be a positive real number. Determine whether there exists a real…

A: |1/(n+1) | &lt; E =&gt; -E &lt; 1/(n+1) &lt; E From the first inequality, we get that,  n+1 &gt;…

Q: Let a and b be any real numbers such that a 0. Prove that a < b.

A:

Q: Prove that, if a, b, and c are real numbers such that a <b< c, then |b| < max{Ja|, Jel}.

A: Consider the given: a, b ,c are real numbers  such that a&lt;b&lt;c

Q: Is every infinite subset of Z x (Z-{0}) a rational number?

A: Given: Infinite subset of ℤ×ℤ-0 To Prove: Every subset is a rational number. Rational Number: Any…

Q: For each part, find the integer 0 ≤ r < |m| such that a ≡ r mod m: (a) m = 7, a = 5 (b) m = 5, a =…

A:

Q: The greatest integer in a real number x, denoted [x], is the largest integer less than or equal to…

A:

Q: For any integer m and any real number x, if x is integer, then x]+[m-x] = m- 1. -. not an

A:

Q: 7. Prove that for each real number x, there is a unique integer n such that n ≤ x ≤ n + 1.

A: