Q. 72 Prove that if ak鈫扡聽and f聽is ... [FREE SOLUTION]

Chapter 7: Q. 72 (page 605)

Prove that if $\{a_{k}\} 鈫� L$ and $f$ is a function that is continuous at $L$ , then $f (a_{k}) 鈫� f (L)$ .

Short Answer

Expert verified

It is proved that if $\{a_{k}\} 鈫� L$ and $f$ is a function that is continuous at $L$ , then $f (a_{k}) 鈫� f (L)$ .

Step by step solution

Step 1. Given Information

The objective is to prove that if $\{a_{k}\} 鈫� L$ and $f$ is a function that is continuous at $L$ , then $f (a_{k}) 鈫� f (L)$ .

Step 2. Prove

As $f$ is a continuous function so for a $蔚 > 0$ there exist $未 > 0$ such as $|f (x) - f (L)| < 蔚鈭赌 x 鈭� N (L, 未)$ .

Now, $\lim_{k 鈫� 鈭�} a_{k} = L$ there exists a $m 鈭� N$ such that

$|a_{n} - L| < 未鈭赌 n 鈮� m$ .

So, for all $n 鈮� m$ , $a_{n} 鈭� N (L, 未)$ .

So, $|f (a_{n}) - f (L)| < 蔚$ for all $n 鈮� m$ .

Hence it is proved that $f (a_{k}) 鈫� f (L)$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Prove that if $\{a_{k}\} 鈫� L$ and $f$ is a function that is continuous at $L$ , then $f (a_{k}) 鈫� f (L)$ .

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Prove

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Theoretical and Mathematical Physics

Calculus

Logic and Functions

Applied Mathematics

Statistics

Study anywhere. Anytime. Across all devices.

91影视

Prove that if ak鈫�Land fis a function that is continuous at L, then fak鈫�fL.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Prove

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Theoretical and Mathematical Physics

Calculus

Logic and Functions

Applied Mathematics

Statistics

Study anywhere. Anytime. Across all devices.

Prove that if $\{a_{k}\} 鈫� L$ and $f$ is a function that is continuous at $L$ , then $f (a_{k}) 鈫� f (L)$ .