Prove the following:

Let a,b ∈R. Suppose a > 0. Prove that there is some n ∈∞ such that b ∈[−na,na].

**Subject Mathematics Real Analysis**

Prove the following:

Let a,b ∈R. Suppose a > 0. Prove that there is some n ∈∞ such that b ∈[−na,na].

Let a,b ∈R. Suppose a > 0. Prove that there is some n ∈∞ such that b ∈[−na,na].

This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.

Let a, b ∈ R. Suppose a > 0. Prove that there is some n ∈ N such that b ∈ [−na, na].

We know there is n ∈ N, b < na , by Theorem 2.67 (Archmidean Property).

Suppose b ≥ 0 we know n ≥ 1 and a > 0 therefore −na < 0 and −na < b then b ∈ [−na, na].

Suppose b < 0: There are two cases:

Case 1: There is n ∈ N such that −na < b then b ∈ [−na, na].

Case 2: b < −na for all n ∈ N...

We know there is n ∈ N, b < na , by Theorem 2.67 (Archmidean Property).

Suppose b ≥ 0 we know n ≥ 1 and a > 0 therefore −na < 0 and −na < b then b ∈ [−na, na].

Suppose b < 0: There are two cases:

Case 1: There is n ∈ N such that −na < b then b ∈ [−na, na].

Case 2: b < −na for all n ∈ N...

This is only a preview of the solution. Please use the purchase button to see the entire solution

Continuity and Differentiability

$200.00

Advances Mathematics

Real Analysis

Relations

Continuous Function

Limits

Differentiable

Contradiction

Theorem

Definition

Points

Neighborhood

Intervals

Advances Mathematics

Real Analysis

Relations

Continuous Function

Limits

Differentiable

Contradiction

Theorem

Definition

Points

Neighborhood

Intervals

Mathematics Questions

$50.00

Mathematics

Analysis

Metrics

Uniform Equicontinuity

Real Values

Functions

Compactness

Points

Homeomorphism

Real Analysis Proofs

$8.00

Real Analysis

Mathematics

Riemann Integrals

Functions

Subintervals

Zero Measures

Discontinuities

Boundaries

Subsets

Points

Processes

Real Analysis

Mathematics

Riemann Integrals

Functions

Subintervals

Zero Measures

Discontinuities

Boundaries

Subsets

Points

Processes

Real Analysis Problem

$10.00

Real Analysis

Intermediate Value Theorem

Closed Bounded Interval

Open Interval

Real Analysis Problem

$10.00

Real Analysis

Bolzano Weierstrass

Sequence

Subsequence

Converge

Diverge

Unbounded Set

Real Analysis Question and It's Solution

$10.00

Mathematics

Real Analysis

Question

Equation

Solution