Proof: Cauchy Sequences are Bounded | Real Analysis

Support the production of this course by joining Wrath of Math to access all my real analysis videos plus the lecture notes at the premium tier!
   / @wrathofmath  
🛍 Get the coolest math clothes in the world! https://mathshion.com/

Real Analysis course:    • Real Analysis  
Real Analysis exercises:    • Real Analysis Exercises  

Get the textbook! https://amzn.to/45kcMjq

We prove that every Cauchy sequence is bounded. We previously discussed what the Cauchy criterion and Cauchy sequences are, and proved that a sequence is Cauchy. We are on our way to proving a sequence converges if and only if it is Cauchy, but to help us in doing that, we will first prove that Cauchy sequences are bounded in today's #realanalysis video lesson.

To do this, we use the definition of Cauchy sequences to then use our useful absolute value inequality equivalence to establish a bound on terms after a certain point. Then, a max and min will clean up the rest.

What are Bounded Sequences:    • What are Bounded Sequences? | Real Analysis  
Proof Convergent Sequence is Bounded:    • Proof: Convergent Sequence is Bounded | Re...  
Proof of the Absolute Value Inequality Equivalence:    • Proof: A Useful Absolute Value Inequality ...  
Intro to Cauchy Sequences:    • Intro to Cauchy Sequences and Cauchy Crite...  
Proof Convergent Sequences are Cauchy:    • Proof: Convergent Sequences are Cauchy | R...  

★DONATE★
◆ Support Wrath of Math on Patreon for early access to new videos and other exclusive benefits:   / wrathofmathlessons  
◆ Donate on PayPal: https://www.paypal.me/wrathofmath

Follow Wrath of Math on...
● Instagram:   / wrathofmathedu  
● Facebook:   / wrathofmath  
● Twitter:   / wrathofmathedu