Let n be a natural number and let y1, Y2, Y3, . . . , Yn be real numbers. Prove by induction that ly1 – Yn| < |y1 – Y2| + |y2 – Y3| + ·.+ \Yn-1 – Yn|- Suppose that f satisfies |f(x)– f(y)| < (y-x)² for all real numbers r and y. Prove that f is a constant function. Hint: Divide the interval from x to y into n equal pieces and apply part a) to y1 Yn = f(y). What does this tell you as n grows very large? f (x) and
Let n be a natural number and let y1, Y2, Y3, . . . , Yn be real numbers. Prove by induction that ly1 – Yn| < |y1 – Y2| + |y2 – Y3| + ·.+ \Yn-1 – Yn|- Suppose that f satisfies |f(x)– f(y)| < (y-x)² for all real numbers r and y. Prove that f is a constant function. Hint: Divide the interval from x to y into n equal pieces and apply part a) to y1 Yn = f(y). What does this tell you as n grows very large? f (x) and
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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![### Exercise
#### (a)
Let \( n \) be a natural number and let \( y_1, y_2, y_3, \ldots, y_n \) be real numbers. Prove by induction that
\[
|y_1 - y_n| \leq |y_1 - y_2| + |y_2 - y_3| + \cdots + |y_{n-1} - y_n|.
\]
#### (b)
Suppose that \( f \) satisfies \( |f(x) - f(y)| \leq (y-x)^2 \) for all real numbers \( x \) and \( y \). Prove that \( f \) is a constant function. Hint: Divide the interval from \( x \) to \( y \) into \( n \) equal pieces and apply part (a) to \( y_1 = f(x) \) and \( y_n = f(y) \). What does this tell you as \( n \) grows very large?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd5936ff2-300d-435d-83da-c517ddce4903%2F9c3ef841-3752-403d-9daf-2ddde7a618de%2Fwrix9sb_processed.png&w=3840&q=75)
Transcribed Image Text:### Exercise
#### (a)
Let \( n \) be a natural number and let \( y_1, y_2, y_3, \ldots, y_n \) be real numbers. Prove by induction that
\[
|y_1 - y_n| \leq |y_1 - y_2| + |y_2 - y_3| + \cdots + |y_{n-1} - y_n|.
\]
#### (b)
Suppose that \( f \) satisfies \( |f(x) - f(y)| \leq (y-x)^2 \) for all real numbers \( x \) and \( y \). Prove that \( f \) is a constant function. Hint: Divide the interval from \( x \) to \( y \) into \( n \) equal pieces and apply part (a) to \( y_1 = f(x) \) and \( y_n = f(y) \). What does this tell you as \( n \) grows very large?
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