Complete sequence spaces and approximate by finite support

Try

A finite screen can show zeros while an infinite sequence still has total absolute sum one. For a positive integer m, put the value 1/m at coordinates m+1 through 2m, and zero everywhere else. There are exactly m nonzero coordinates. Its ℓ1 norm is m·(1/m)=1, its ℓ2 norm is √(m/m²)=1/√m, and its supremum norm is 1/m.

For each fixed coordinate k, the value is zero once m≥k. Thus every coordinate tends to zero, yet convergence in ℓ1 fails. The same sequence does converge to zero in ℓ2 and in the supremum norm. Before speaking of approximation, name the norm: it specifies what kind of error must become small.

Predict

Predict what happens when m=32 but a plot displays only the first 24 coordinates. Every displayed value is zero. Predict also the ℓ1 error of keeping only the first 16 coordinates: it is still one, because all 32 nonzero values lie at coordinates 33 through 64.

Does this refute completeness of ℓ1? No. Completeness promises a limit for norm-Cauchy sequences. The blocks for m and 2m have disjoint supports, so their ℓ1 distance is 2. They are not ℓ1 Cauchy. A valid counterexample to completeness must satisfy the entire Cauchy hypothesis, not just coordinatewise convergence.

Experiment

Can a finite prefix hide the norm?

Only coordinates 1–24 are plotted. The full norm and truncation error below use the stated infinite sequence formula, not an extrapolation from those points. P_K keeps coordinates 1,…,K and replaces all later coordinates by zero.

Sequence familyGeometric: xₖ=2⁻ᵏConstant: xₖ=1Moving unit vector eₘEscaping block: height 1/m Normℓ1 sum normℓ2 square-sum normℓ∞ supremum norm Retained coordinates K124816 Family index m12481632 Reset

xₖ=2⁻ᵏ for every k≥1. Geometric-series identities give the exact infinite sums.

Numerical values are rounded. The tail formulas describe the entire sequence, including coordinates outside the plot.

• Norm of displayed coordinates 1–24: 0.9999999.
• Full sequence norm: 1.
• Truncation error ‖x−P_Kx‖: 0.0625. Formula: 2^(−K).

Nonzero coordinates exist beyond this plot. A zero displayed prefix does not imply a zero sequence or zero norm.

Inspect the displayed coordinates

Finite prefix only; decimals rounded

k	xₖ	P_Kxₖ
1	0.5	0.5
2	0.25	0.25
3	0.125	0.125
4	0.0625	0.0625
5	0.03125	0
6	0.015625	0
7	0.0078125	0
8	0.00390625	0
9	0.001953125	0
10	0.0009765625	0
11	0.0004882813	0
12	0.0002441406	0
13	0.0001220703	0
14	0.00006103516	0
15	0.00003051758	0
16	0.00001525879	0
17	0.000007629395	0
18	0.000003814697	0
19	0.000001907349	0
20	9.536743e-7	0
21	4.768372e-7	0
22	2.384186e-7	0
23	1.192093e-7	0
24	5.960464e-8	0

Try the escaping block with m=32. Its displayed prefix is zero, but its ℓ1 norm remains 1. Compare ℓ2 and ℓ∞, then explain the limit as m grows using the formulas. For each fixed coordinate k the block eventually leaves k; that alone does not prove convergence in every norm. Controls are temporary and award no completion.

Compare the escaping block in the three available norms. Increase m while keeping the truncation coordinate K fixed. Record the analytic full norm and the analytic truncation error, even when the visible prefix is empty. The finite graph is evidence about its displayed coordinates; the formulas supply the tail information.

Then choose the geometric sequence xₖ=2⁻ᵏ, indexed by k≥1. Predict its ℓ1 truncation error at K=4 before reading the displayed calculation. It is Σ from k=5 onward of 2⁻ᵏ=1/16. The ℓ2 error is 2⁻ᴷ/√3, and the supremum error is 2⁻⁽ᴷ⁺¹⁾. The different errors arise from the same omitted coordinates.

Changing the selected norm does not change the sequence. It changes which differences are considered small. The constant-one sequence belongs to ℓ∞ but not to any finite ℓp.

Understand

For 1≤p<∞, ℓp consists of real sequences x with Σ|xₖ|ᵖ finite, and its norm is the pth root of that sum. The preceding lesson proves the triangle inequality. A nonnegative infinite sum means the supremum of its finite partial sums. ℓ∞ consists of bounded sequences with norm supₖ|xₖ|. Unlike the rational-to-real construction, we may now use established real completeness for scalar coordinates.

Construct an ℓp limit without interchanging an infinite sum and a limit. Suppose x⁽ⁿ⁾ is norm Cauchy. For every k,

|x⁽ⁿ⁾ₖ−x⁽ᵐ⁾ₖ|≤||x⁽ⁿ⁾−x⁽ᵐ⁾||p.

Thus each coordinate is a scalar Cauchy sequence and has a limit xₖ. First prove that the resulting x belongs to the space. A norm-Cauchy sequence is norm bounded: fix a tail term within distance 1 of every later term, then include the finite prefix in one bound B. For every fixed integer J, finite-sum continuity gives

Σ over k≤J of |xₖ|ᵖ = lim as n grows of Σ over k≤J of |x⁽ⁿ⁾ₖ|ᵖ ≤ Bᵖ.

Taking the supremum over J proves Σ|xₖ|ᵖ≤Bᵖ. Membership has now been established. Given ε>0, choose N so that ||x⁽ⁿ⁾−x⁽ᵐ⁾||p<ε/2 for n,m≥N. Fix n≥N and a finite J; pass m to infinity only in the finite sum of differences. It is at most (ε/2)ᵖ. Taking the supremum over J gives ||x⁽ⁿ⁾−x||p≤ε/2<ε. Hence ℓp is complete. The order of “finite limit, then supremum” is the proof mechanism, not a cosmetic detail.

The supremum case has a uniform coordinate bound. A Cauchy sequence in ℓ∞ again has a common bound B and coordinate limits xₖ, so |xₖ|≤B for every k. For n,m≥N the Cauchy estimate controls every coordinate by ε/2 with the same N. Fixing n and k and passing m to infinity gives |x⁽ⁿ⁾ₖ−xₖ|≤ε/2. Taking the supremum over k proves norm convergence. Coordinatewise convergence without that common estimate would not suffice: the moving unit vectors give the counterexample.

Identify a closed subspace. Let c0 be the sequences whose coordinates tend to zero. Linear combinations stay in c0. If x⁽ⁿ⁾∈c0 and x⁽ⁿ⁾ tends uniformly to x, choose one n with uniform error below ε/2. Then choose K so that the coordinates of this fixed x⁽ⁿ⁾ beyond K are smaller than ε/2. Their sums show |xₖ|<ε for k>K. Thus x∈c0. This proves c0 closed in complete ℓ∞ and therefore complete. Notice the two choices occur in order: first a single approximating sequence, then its tail cutoff.

Identify what finite support approximates. Let c00 denote sequences with finitely many nonzero entries, and P_Kx the truncation retaining coordinates 1 through K. For finite p,

||x−P_Kx||pᵖ = Σ over k>K of |xₖ|ᵖ,

which tends to zero for x∈ℓp. For x∈c0 the corresponding supremum error is sup over k>K of |xₖ|, also tending to zero. Thus c00 is dense in ℓp and in c0 with their stated norms. Since c0 is closed in ℓ∞ and contains c00, the supremum-norm closure of c00 is exactly c0.

The constant-one sequence has distance exactly 1 from either c00 or c0. Every finitely supported approximation leaves a tail of differences equal to 1. For y∈c0, |1−yₖ| tends to 1, so its supremum is at least 1. In both cases y=0 attains that lower bound. This is a quantitative obstruction to density in ℓ∞, not merely a failed choice of truncation.

Apply

For x⁽ⁿ⁾ and x in a fixed finite ℓp, coordinate convergence becomes norm convergence precisely when the late sequences have uniformly small tails: for every ε>0, some K,N give ||x⁽ⁿ⁾−P_Kx⁽ⁿ⁾||p<ε for all n≥N. Prove sufficiency by splitting the error into the sequence tail, the first K coordinate differences, and the limit tail. Choose one K large enough for both tail requirements; coordinate convergence controls the remaining finite vector. Prove necessity using norm convergence and the tail of x. The written task supplies the full quantified argument.

For a fresh calculation take xₖ=3⁻ᵏ. Its ℓ1 norm is 1/2, and its ℓ1 truncation error is 3⁻ᴷ/2. Its supremum error is 3⁻⁽ᴷ⁺¹⁾. Explain why checking only the first K coordinates would give zero error for the truncation there, while missing these nonzero whole-sequence errors.

Continue

Return to complete metric subspaces for the closed-subspace argument. The broader Lp completeness proof generalizes this setting to measurable functions and almost-everywhere equivalence. It is a subsequent connection, not a prerequisite for the elementary coordinate proof here.

Keep membership, convergence and density as separate claims in your evidence. A table can check an exact finite prefix; an analytic tail estimate controls what the table omits. Completeness guarantees a limit under a norm-Cauchy hypothesis, and does not turn every coordinatewise experiment into norm convergence.