Learn with Amar
Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Verify the Cauchy property independently of naming a limit.
- Justify the conclusion "m,n≥N implies |aₘ−aₙ|<ε" using the stated assumptions.
Before you start
Sequences and absolute-value bounds.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Show that aₙ=1/n is Cauchy in R.
Why this math matters
Verify the Cauchy property independently of naming a limit. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The metric is ordinary absolute distance.
- Indices are positive integers.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Check convergence through distances between late terms
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Show that aₙ=1/n is Cauchy in R.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
For m,n≥N, |1/m−1/n|≤1/m+1/n≤2/N
A simple common bound controls every pair of late terms.
Work through the mathematics
Choose N>2/ε
The bound becomes smaller than the requested tolerance.
Check the conclusion
m,n≥N implies |aₘ−aₙ|<ε
This is the Cauchy property; completeness of R then guarantees a real limit.
The result
m,n≥N implies |aₘ−aₙ|<ε
This is the Cauchy property; completeness of R then guarantees a real limit.
Common mistakes to catch
- Cauchy controls all late pairs, not only neighbors.
- Completeness is a property of the ambient space.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Does the same argument need m and n to be consecutive?
Show a hint
Inspect the bound.
Reveal answer and explanation
No
It works for every pair beyond N, which is what Cauchy requires.
Practice 2
Is small successive difference alone enough for convergence?
Show a hint
Consider partial sums of 1/n.
Reveal answer and explanation
No
Harmonic partial sums have increments tending to zero but do not converge.
Take the idea with you
Explain why a solver's tiny last step alone may not establish convergence.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Use an invariant interval to prove a recurrence converges
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