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Teaching video
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Graduate chapters and video availability01 · Read and understand
What you will learn
- Check a contraction constant and derive a computable error bound.
- Justify the conclusion "|xₙ−1|≤3⁻ⁿ|x₀−1|" using the stated assumptions.
Before you start
Complete metric spaces and geometric series.
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
For T(x)=(x+2)/3 on R, prove convergence of xₙ₊₁=T(xₙ) to its fixed point.
Why this math matters
Check a contraction constant and derive a computable error bound. 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 iteration is performed on all of R.
- The contraction constant is uniformly below one.
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.
Turn a contraction into a certified iterative solver
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For T(x)=(x+2)/3 on R, prove convergence of xₙ₊₁=T(xₙ) to its fixed point.
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
|T(x)−T(y)|=|x−y|/3
The map is a contraction with constant q=1/3 on a complete space.
Work through the mathematics
x*=T(x*) gives x*=1
Solve the fixed-point equation to identify the unique candidate.
Check the conclusion
|xₙ−1|≤3⁻ⁿ|x₀−1|
Repeated contraction proves convergence and gives an explicit a priori error bound.
The result
|xₙ−1|≤3⁻ⁿ|x₀−1|
Repeated contraction proves convergence and gives an explicit a priori error bound.
Common mistakes to catch
- Having a fixed point alone does not imply every iteration converges.
- A local derivative estimate is not automatically a global contraction bound.
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
What a posteriori bound uses the last step?
Show a hint
Sum the remaining geometric increments.
Reveal answer and explanation
|xₙ−x*|≤q|xₙ−xₙ₋₁|/(1−q)
Future increments are bounded by successive powers of q.
Practice 2
Why include completeness?
Show a hint
A Cauchy iteration needs a limit in the space.
Reveal answer and explanation
The limit might otherwise lie outside the space
A contraction on a noncomplete domain need not have a fixed point there.
Take the idea with you
Convert an iteration's contraction estimate into a stopping criterion.
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: Differentiate a corner through integration against tests
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