Lecture 6: The Uncountabality of the Real Numbers

TL;DR

The triangle inequality, convergence of sequences, and the limit theorem are fundamental concepts in real analysis.

Transcript

[SQUEAKING] [RUSTLING] [CLICKING] CASEY RODRIGUEZ: So we ended last time with these elementary properties that you know about the absolute value. So let me prove one more theorem about the absolute value, which maybe you didn't cover in calculus. But it's of fundamental importance in real analysis, which is the triangle inequality, which states tha... Read More

Key Insights

• 🔺 The triangle inequality is a fundamental property in real analysis and relates to side lengths in a triangle.
• 💗 Bounded sequences do not grow arbitrarily large and have an upper bound on their elements.
• 🍉 The limit of a sequence represents the value that the terms of the sequence converge to as n approaches infinity.

Questions & Answers

Q: What is the triangle inequality and why is it important?

The triangle inequality states that the absolute value of the sum of two real numbers is less than or equal to the sum of the absolute values of the numbers. It is important because it establishes a relationship between the lengths of sides in a triangle and is a fundamental property in real analysis.

Q: What does it mean for a sequence to be bounded?

A sequence is bounded if there is a real number b such that all elements of the sequence are less than or equal to b in absolute value. This means that the elements of the sequence do not grow arbitrarily large.

Q: What is the definition of a limit of a sequence?

The limit of a sequence x sub n is a real number x if for every epsilon positive, there exists a natural number capital M such that for all n greater than or equal to capital M, the absolute value of x sub n minus x is less than epsilon. This means that as the terms of the sequence progress, they get arbitrarily close to x.

Q: Why is it not possible for a convergent sequence to have multiple limits?

Suppose a sequence converges to two different limits, x and y. This would mean that for every positive epsilon, there exists a natural number capital M such that for all n greater than or equal to capital M, the terms of the sequence are close to both x and y. However, by the theorem that states the absolute value of x minus y is less than epsilon, it is not possible for x and y to be different. Therefore, a convergent sequence can only have one limit.

Summary & Key Takeaways

• The triangle inequality states that for all xy in R, the absolute value of x plus y is less than or equal to the absolute value of x plus the absolute value of y.

• A sequence of real numbers is bounded if there exists a real number b such that all elements of the sequence are less than or equal to b in absolute value.

• The limit of a sequence x sub n is a real number x if for every epsilon positive, there exists a natural number capital M such that for all n greater than or equal to capital M, the absolute value of x sub n minus x is less than epsilon.