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Exercise 2.3.2. Using only Definition 2.2.3, prove that if (rm) → 2, then (a) (2) → 1; (b) (1/r,) → 1/2. (For this exercise the Algebraic Limit Theorem is off-limits, so to speak.)

Exercise 2.3.2. Using only Definition 2.2.3, prove that if (rm) → 2, then (a) (2) → 1; (b) (1/r,) → 1/2. (For this exercise the Algebraic Limit Theorem is off-limits, so to speak.)

Advanced Engineering Mathematics
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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Definition 2.2.3 (Convergence of a Sequence). A sequence (an) converges to a real number a if, for every positive number e, there exists an NeN such that whenever n > N it follows that lan – a| < e. To indicate that (an) converges to a, we usually write either lim an = a or (an) → a. The notation lim,,-+ a, = a is also standard. In an effort to decipher this complicated definition, it helps first to consider the ending phrase "lan – al < e," and think about the points that satisfy an inequality of this type.
Transcribed Image Text:Definition 2.2.3 (Convergence of a Sequence). A sequence (an) converges to a real number a if, for every positive number e, there exists an NeN such that whenever n > N it follows that lan – a| < e. To indicate that (an) converges to a, we usually write either lim an = a or (an) → a. The notation lim,,-+ a, = a is also standard. In an effort to decipher this complicated definition, it helps first to consider the ending phrase "lan – al < e," and think about the points that satisfy an inequality of this type.
Exercise 2.3.2. Using only Definition 2.2.3, prove that if (xm) → 2, then (a) (2) → 1; (b) (1/r,) → 1/2. (For this exercise the Algebraic Limit Theorem is off-limits, so to speak.)
Transcribed Image Text:Exercise 2.3.2. Using only Definition 2.2.3, prove that if (xm) → 2, then (a) (2) → 1; (b) (1/r,) → 1/2. (For this exercise the Algebraic Limit Theorem is off-limits, so to speak.)
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