Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Compare pointwise behavior with a worst-case error over a whole interval.
- Justify the conclusion "sup|fₙ−f|=1, so convergence is not uniform" using the stated assumptions.
Before you start
Functions and supremum norms.
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
Analyze fₙ(x)=xⁿ on [0,1].
Why this math matters
Compare pointwise behavior with a worst-case error over a whole interval. 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 main domain includes the endpoint one.
- The supremum need not be attained.
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.
Watch a sequence converge pointwise but not uniformly
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Analyze fₙ(x)=xⁿ on [0,1].
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 x<1, xⁿ→0; at x=1, xⁿ=1
Each fixed point has a well-defined limiting value.
Work through the mathematics
The limit f is zero on [0,1) and one at 1
A boundary jump appears even though every fₙ is continuous.
Check the conclusion
sup|fₙ−f|=1, so convergence is not uniform
Values just below one keep the worst-case error arbitrarily near one for every finite n.
The result
sup|fₙ−f|=1, so convergence is not uniform
Values just below one keep the worst-case error arbitrarily near one for every finite n.
Common mistakes to catch
- An input held fixed differs from an input chosen after n.
- Changing the domain can change uniform convergence.
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
Is convergence uniform on [0,1/2]?
Show a hint
Bound xⁿ by (1/2)ⁿ.
Reveal answer and explanation
Yes
The uniform error is at most 2⁻ⁿ and tends to zero.
Practice 2
Why does the discontinuous limit warn against uniform convergence?
Show a hint
A uniform limit of continuous functions is continuous.
Reveal answer and explanation
It contradicts that preservation theorem
Pointwise convergence alone does not preserve continuity.
Take the idea with you
Distinguish average or pointwise numerical agreement from a uniform error guarantee.
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: Find a continuous function with a corner
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