Chapter 7: Q. 75 (page 605)

Prove Theorem 7.9. That is, let $a : [1, \infty) \to ℝ$ be a continuous function and let $a_{k} = a (k)$ for every $k \in ℤ^{+}$ . Show that if $\lim_{x \to \infty} a (x) = L$ , then $\{a_{k}\} \to L$

Short Answer

Expert verified

The theorem is hence proved.

Step by step solution

Step 1. Given Information.

The objective is to show that if $\lim_{x \to \infty} a (x) = L$ then $\{a_{k}\} \to L$ .

Step 2. Forming the equation.

It is given that $\lim_{x \to \infty} a (x) = L$ , therefore, for given, $ε > 0$ , there exists a positive integer such that

$|a (x) - L| < ε f o r x > N . . . . . . . (1)$

Also, $a (k) = a_{k}$

Therefore, using equation (1) we get,

$|a (k) - L| < ε f o r k > N . . . . . . (2) |a_{k} - L| < ε f o r k > N$

Step 3. Proving the theorem.

Thus for given $ε > 0$ , there exists a positive integer $N$ such that

$|a_{k} - L| < ε f o r k > N$

Hence, $\{a_{k}\} \to L$ .

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Prove Theorem 7.9. That is, let $a : [1, \infty) \to ℝ$ be a continuous function and let $a_{k} = a (k)$ for every $k \in ℤ^{+}$ . Show that if $\lim_{x \to \infty} a (x) = L$ , then $\{a_{k}\} \to L$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Forming the equation.

Step 3. Proving the theorem.

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Prove Theorem 7.9. That is, let a:[1,∞)→ℝbe a continuous function and let ak=akfor every k∈ℤ+. Show that if limx→∞a(x)=L, then ak→L

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Forming the equation.

Step 3. Proving the theorem.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Geometry

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Probability and Statistics

Study anywhere. Anytime. Across all devices.

Prove Theorem 7.9. That is, let $a : [1, \infty) \to ℝ$ be a continuous function and let $a_{k} = a (k)$ for every $k \in ℤ^{+}$ . Show that if $\lim_{x \to \infty} a (x) = L$ , then $\{a_{k}\} \to L$