The sequence b_n=1−1/n increases and stays below 1. Its limit is suggested by its formula, but the monotone convergence theorem works even when no formula for the limit is available. It says an increasing real sequence bounded above converges. Each hypothesis matters: n increases without a finite limit, while (−1)^n is bounded without converging. The theorem combines direction of motion with a global barrier.
Predict
Predict what happens if a bounded sequence oscillates. Convergence of the whole sequence may fail, but selected terms can still approach a point. A subsequence is formed by strictly increasing indices n₁<n₂<⋯; it preserves order and goes arbitrarily far into the original sequence. Selecting one convenient term repeatedly is not a subsequence. Distinguish an infinite extraction from merely deleting a few initial values.
Experiment
Compare three cases.
Sequence
Bounded?
Convergent?
Useful subsequence
1−1/n
Yes
Yes
Every subsequence has limit 1
(−1)^n
Yes
No
Even indices give 1
n
No
No
Every subsequence is unbounded
Explain why a finite plotted oscillation cannot establish failure of convergence. The parity argument does: both separated values occur beyond every cutoff. Then construct a bounded sequence with infinitely many repeated values and describe an increasing-index extraction of one of them.
Understand
For an increasing bounded-above sequence, let L=sup a_n. Given ε>0, the supremum criterion supplies an index N with a_N>L−ε. Monotonicity gives L−ε<a_N≤a_n≤L for every n≥N, proving convergence. This is the exact point where completeness creates the candidate limit.
Bolzano–Weierstrass says every bounded real sequence has a convergent subsequence. Place all its terms in a closed bounded interval. Bisect and retain a closed half containing infinitely many indexed terms; at least one half must qualify. Repeat. Nested intervals with lengths tending to zero have one common point L. Choose n_k larger than n_(k−1) with a_(n_k) inside the kth interval. Infinitely many eligible indices guarantee the choice. Both the selected term and L lie in that interval, so their distance tends to zero.
Apply
Repair “Bolzano–Weierstrass proves every bounded sequence converges.” The conclusion concerns an extracted subsequence. The alternating sequence remains a counterexample to full convergence. Conversely, if the original sequence does converge, every subsequence shares its limit: strictly increasing positive indices satisfy n_k≥k, so an original tail estimate also controls a sufficiently late subsequence tail.
The theorem is special to the complete real setting. A bounded rational sequence approaching √2 has no subsequence converging to a rational limit, although it has a real limit. Always say where convergence is required and where the extracted limit belongs.
A recurrence needs a domain proof
Start at x₀=4 and repeatedly compute xₙ₊₁=4−4/xₙ. The first values are 4, 3, 8/3 and 5/2. Solving the fixed-point equation suggests two, but does not prove the divisions remain defined or that the sequence converges. First predict what changes if the initial value is one.
Prove the barrier before taking a limit
Rule: xₙ₊₁=2r−r²/xₙ. Predict whether the denominator stays positive. All variables are dimensionless; at most sixteen exact rational transitions are computed.
8 transitions computed. This is a bounded prefix, not an infinite convergence test.
The symbolic invariant xₙ≥r proves every future denominator positive and the limit r=2. The conclusion comes from the proof, not the plot.
Index n
Exact xₙ
Exact error xₙ−r
Next denominator allowed?
0
4
2
Yes: nonzero
1
3
1
Yes: nonzero
2
8/3
2/3
Yes: nonzero
3
5/2
1/2
Yes: nonzero
4
12/5
2/5
Yes: nonzero
5
7/3
1/3
Yes: nonzero
6
16/7
2/7
Yes: nonzero
7
9/4
1/4
Yes: nonzero
8
20/9
2/9
Yes: nonzero
Connecting segments guide the eye; only integer-index dots are sequence terms. Temporary controls award no mastery.
This develops the invariant-recurrence demand in MIT 18.100B problem set 3, question 4. Our indexing starts at zero; its first term is indexed one. The surrounding contraction problem is a separate theorem, not a justification for this example.
Establish the barrier and direction
For r>0, write T(x)=2r−r²/x. If x≥r, the denominator is positive and
Thus induction from x₀≥r proves every term exists and lies in [r,x₀], and the sequence is nonincreasing. If x₀=r it is constant; if x₀>r, the first identity preserves strict inequality and the second gives strict decrease. Monotone convergence now supplies a limit L≥r>0. Only at this stage may limit laws pass through the reciprocal. They give L=2r−r²/L, hence (L−r)²=0 and L=r.
A changed initial value can invalidate the entire argument. Starting at r/2 gives the next term zero, so the following division is undefined. Starting at 3r/4 gives 2r/3, then r/2, then zero. The fixed-point equation is unchanged, but these sequences do not continue indefinitely. An invalid iterate must not be silently replaced with the candidate limit.
Derive the rate instead of naming it a contraction
When x₀>r, let eₙ=xₙ−r>0. The first identity gives eₙ₊₁=r eₙ/(r+eₙ). Taking reciprocals yields
1/eₙ₊₁ = 1/eₙ + 1/r, so eₙ = r e₀/(r+n e₀).
For x₀=2r this reduces to eₙ=r/(n+1). With r=2, the exact terms are 2+2/(n+1); this proves the limit and quantifies the number of iterations needed for a tolerance. The bounded trace checks arithmetic, while induction and the formula establish every index.
There is no uniform contraction factor q<1 on [r,2r]. For x>r, the ratio |T(x)−T(r)|/|x−r| is r/x, tending to one as x decreases to r. Every proposed q<1 is exceeded by some such pair. The errors decay on the order of 1/n, although each positive error shrinks. Convergent monotone iteration and strict contraction are different sufficient mechanisms; choosing the right theorem is part of the proof.
Continue
Optional depth studies cluster points, the limits of subsequences. A bounded sequence may have one, several or infinitely many such points. The set of cluster points records behavior that a single average or finite trace cannot capture.
Write the bisection proof with two separate choices: the next interval and the next index. Omitting the increasing-index condition can invalidate the extraction. Next, a Cauchy condition will provide the extra control needed to turn one convergent subsequence into convergence of the entire original sequence, revealing another equivalent expression of real completeness.
Apply · explain · revise
Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
For r>0 and x₀≥r, prove xₙ₊₁=2r−r²/xₙ exists for every n and tends to r. Then check x₀=3r/4. Why is the fixed-point equation alone insufficient?
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For the preceding recurrence with x₀=2r, derive the exact error and prove no uniform contraction factor below one works on [r,2r].
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
What is the limit of the even-index subsequence of a_n=(−1)^n+1/n?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Every bounded real sequence:
Prove an increasing bounded-above sequence converges to the supremum of its values.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Prove every subsequence of a convergent sequence has the same limit.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Prove every bounded real sequence has a convergent subsequence using nested bisection.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Explain why a_n=(−1)^n+1/n has no limit but has two subsequential limits.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
