Learn with Amar
Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Combine monotonicity and boundedness before solving a fixed-point equation.
- Justify the conclusion "aₙ→L and L²=2+L, so L=2" using the stated assumptions.
Before you start
Sequences, inequalities, and continuity.
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
Let a₁=1 and aₙ₊₁=√(2+aₙ). Prove convergence and identify the limit.
Why this math matters
Combine monotonicity and boundedness before solving a fixed-point equation. 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:
- Square roots denote the nonnegative root.
- The initial value is one in the main argument.
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.
Use an invariant interval to prove a recurrence converges
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Let a₁=1 and aₙ₊₁=√(2+aₙ). Prove convergence and identify the limit.
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
1≤aₙ≤2 implies 1≤√(2+aₙ)≤2
The interval is preserved by the recurrence.
Work through the mathematics
For 1≤x≤2, √(2+x)≥x because 2+x−x²≥0
The sequence is nondecreasing while staying bounded.
Check the conclusion
aₙ→L and L²=2+L, so L=2
Monotone convergence establishes existence first; the allowed interval excludes the other root −1.
The result
aₙ→L and L²=2+L, so L=2
Monotone convergence establishes existence first; the allowed interval excludes the other root −1.
Common mistakes to catch
- A candidate limit is not proof that a limit exists.
- Reject algebraic roots incompatible with the sequence's range.
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
Why is solving L²=L+2 alone not a convergence proof?
Show a hint
A recurrence may fail to approach a fixed point.
Reveal answer and explanation
It only finds possible limits
Boundedness and monotonicity supplied the missing existence argument.
Practice 2
What happens with a₁=2?
Show a hint
Substitute into the recurrence.
Reveal answer and explanation
The sequence stays two
Two is already a fixed point.
Take the idea with you
Find an invariant interval before iterating another nonlinear update.
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: Detect nonconvergence through incompatible subsequences
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