Question
upstudy study bank question image url

Use the formal definition of the limit of a sequence to prove the following limit. \( \lim _{n \rightarrow \infty}\left(\frac{1}{n}\right)=0 \) Complete the steps of the proof below. To show that \( \lim _{n \rightarrow \infty}\left(\frac{1}{n}\right)=0 \), for every \( \varepsilon>0 \), show that there exists an integer \( N \) such that for all if \( n>N \), then \( |\square-0|<\varepsilon \).

Ask by O'Quinn Campbell. in the United States
Mar 15,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

**Proof:** For any \( \varepsilon > 0 \), choose \( N = \left\lceil \frac{1}{\varepsilon} \right\rceil \). Then, for all \( n > N \), \[ \frac{1}{n} < \varepsilon. \] Thus, \[ \lim_{n \to \infty} \frac{1}{n} = 0. \]

Solution

Sign in to Unlock Answers for Free!

A Learning Platform Trusted by Millions of Real Students and Teachers.

star-icon Unlock

Answered by UpStudy AI and reviewed by a Professional Tutor

Extra Insights

To show that \( \lim _{n \rightarrow \infty}\left(\frac{1}{n}\right)=0 \), for every \( \varepsilon>0 \), we need to find an integer \( N \) such that for all \( n>N \), it follows that \( |\frac{1}{n}-0|<\varepsilon \). We start with the inequality: \[ |\frac{1}{n}-0| = \frac{1}{n} < \varepsilon. \] To solve for \( n \), we can rearrange this inequality: \[ \frac{1}{n} < \varepsilon \implies n > \frac{1}{\varepsilon}. \] Now, we can choose \( N = \lceil \frac{1}{\varepsilon} \rceil \), the smallest integer greater than or equal to \( \frac{1}{\varepsilon} \). Therefore, for all integers \( n > N \): \[ n > \frac{1}{\varepsilon} \implies \frac{1}{n} < \varepsilon. \] This shows that \( |\frac{1}{n}-0|<\varepsilon \) holds, thus proving: \[ \lim _{n \rightarrow \infty}\left(\frac{1}{n}\right)=0. \] This completes our proof! In conclusion, we have shown that for any positive \( \varepsilon \), there exists an integer \( N \) such that for all \( n > N \), the condition holds true, satisfying the formal definition of the limit of a sequence.

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy