Prove the following statement. For every real number x, if x − [x] ≥ 1/1/12 then [2x] = 2[x] + 1. Proof: Suppose x is any real number such that X- <- [x] ²/1/201 2[x] ≥ 1 or, equivalently, -Select--- Multiply both sides of the inequality by 2 to obtain 2x Now by definition of floor, x < [x] + 1, and hence ---Select--- (**). Put inequalities (*) and (**) together to obtain 2[x] + 1 ? 2x ? ✓ ---Select--- Thus, by definition of floor, [2x] = 2[x] + 1.
Prove the following statement. For every real number x, if x − [x] ≥ 1/1/12 then [2x] = 2[x] + 1. Proof: Suppose x is any real number such that X- <- [x] ²/1/201 2[x] ≥ 1 or, equivalently, -Select--- Multiply both sides of the inequality by 2 to obtain 2x Now by definition of floor, x < [x] + 1, and hence ---Select--- (**). Put inequalities (*) and (**) together to obtain 2[x] + 1 ? 2x ? ✓ ---Select--- Thus, by definition of floor, [2x] = 2[x] + 1.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Prove the following statement.
For every real number x, if x − [x] >
²1/1/1/1₁
Proof: Suppose x is any real number such that
then [2x]
2[x] + 1.
X- [x] = 1/2.
Multiply both sides of the inequality by 2 to obtain 2x - 2[x] ≥ 1 or, equivalently, --Select---
Now by definition of floor, x < [x] + 1, and hence ---Select---
(**).
Put inequalities (*) and (**) together to obtain 2[x] + 1 ? ✓ 2x ? ✓ ---Select--- ✓
Thus, by definition of floor, [2x] = 2[x] + 1.
(*).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff3cf874b-7a7b-478f-a7b8-421442e72224%2Fbca47d85-38a9-4557-8db7-5f5c2ce25bf2%2F7ju4pjw_processed.png&w=3840&q=75)
Transcribed Image Text:Prove the following statement.
For every real number x, if x − [x] >
²1/1/1/1₁
Proof: Suppose x is any real number such that
then [2x]
2[x] + 1.
X- [x] = 1/2.
Multiply both sides of the inequality by 2 to obtain 2x - 2[x] ≥ 1 or, equivalently, --Select---
Now by definition of floor, x < [x] + 1, and hence ---Select---
(**).
Put inequalities (*) and (**) together to obtain 2[x] + 1 ? ✓ 2x ? ✓ ---Select--- ✓
Thus, by definition of floor, [2x] = 2[x] + 1.
(*).
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