Skip to content
Menu
中文Sign in
← All questions

What does the green dashed line indicate about the relationship between the supremum L and an arbitrary epsilon?

The green dashed line indicates the value L−ϵL - \epsilon for any given ϵ>0\epsilon > 0. It visually demonstrates the concept of an 'epsilon neighborhood' below the supremum. Its purpose is to show that although LL is the least upper bound, we can always find a margin below it (L−ϵL - \epsilon) that the sequence eventually surpasses.

Conditions

  • ϵ>0\epsilon > 0 is an arbitrary positive distance.
  • LL is the supremum.

Reasoning, step by step

  1. Locate the green dashed line below the red line.
  2. Identify its label as L−ϵL - \epsilon.
  3. Understand that ϵ\epsilon represents the vertical gap between the red and green lines.
  4. Recognize that the sequence terms will eventually rise above this green line.

Example

The script notes: 'Below it, a green dashed line indicates L - ε for any given ε>0ε > 0.'

Common misconceptions

  • Thinking the green line is a fixed constant independent of epsilon.
  • Believing the sequence stops at the green line; it passes through it towards L.

Watch the explanation

0:13 – 0:24Watch this moment ↗

Explore next

Answers are generated from source material and independently checked. Consult the original video or creator if something is unclear.