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Chapter 7: Q. 85 (page 605)

Prove that every convergent sequence is bounded.

Short Answer

Expert verified

Proved that every convergent sequence is bounded

Step by step solution

01

Step 1. Given information

Let consider the given convergent sequence $\{a_{k}\}$ converging to limitL.

02

Step 2. Prove that the convergent sequence is bounded

The strategy is to prove that the sequence is bounded,show that there exits a positive integer Msuch that $|a_{k}| \leq M f o r a l l k .$

The sequence $\{a_{k}\}$ is convergent and converges to L

By the defination of convergence,for $ε = 1 (> 0)$ there is a positive integer N,such that

$|a_{k} - L| < ε f o r k \geq N |a_{k} - L| < 1 f o r k \geq N$

for all $k \geq N,$ fix Nsuch that

$|a_{k}| \leq |a_{k} - L| + |L| |a_{k}| < 1 + |L| (B e c a u s e |a_{k} - L| < 1)$

For all $k \geq N$ ,choose $M = m a x \{|a_{1}|, |a_{2}|, |a_{3}|, . . . . . . . . . . . . |a_{N}|, 1 + |L|\}$

Therefore, $|a_{k}| < M f o r k \geq M$

Thus the sequence $\{a_{k}\}$ is bounded

Therefore,every convergent sequence is bounded

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Most popular questions from this chapter

Find an example of a continuous function f : $[1, \infty) \to R$ such that $\int_{1}^{\infty} f (x) d x$ diverges and localid="1649077247585" $\sum_{k = 1}^{\infty} f (k)$ converges.

Let f(x) be a function that is continuous, positive, and decreasing on the interval $[1, \infty)$ such that $\underset{x \to \infty}{l i m} f (x) = α > 0$ , What can the integral tells us about the series $\sum_{k = 1}^{\infty} f (k)$ ?

What is meant by the remainder $R_{n}$ of a series $\sum_{k = 1}^{\infty} a_{k}$

Let $\sum_{k = 1}^{\infty} a_{k} b e a c o n v e r g e n t s e r i e s a n d \sum_{k = 1}^{\infty} b_{k} b e a d i v e r g e n t s e r i e s .$ Prove that the series $\sum_{k = 1}^{\infty} (a_{k} + b_{k})$ diverges.

Express each of the repeating decimals in Exercises 71–78 as a geometric series and as the quotient of two integers reduced to lowest terms.

$0.6345345 . . .$

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