Bolzano–Weierstrass Explorer
Extract convergent subsequences from bounded sequences. Try
Monotone
(LIS/LDS) or
Nested intervals
.
Sequence (xₙ)
Harmonic — xₙ = 1/n (convergent)
Alternating — xₙ = (−1)ⁿ (bounded, divergent)
Sine — xₙ = sin n (bounded, divergent)
Oscillatory — xₙ = 1 + (−1)ⁿ/n (convergent)
Fractional — xₙ = {n√2} (bounded, dense)
Periodic — xₙ = (n mod 5)/5 (bounded)
Custom — xₙ = sin(n)/n + (−1)ⁿ/√n
Custom xₙ
Number of terms N
Method
Monotone
Nested intervals
ε-band around L̂
Extract subsequence
Zoom to ε-band
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Bounded range
—
Method / type
—
Subsequence length
—
Estimated limit L̂
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$$\textbf{Bolzano–Weierstrass }(\mathbb{R})\\ \text{Every bounded sequence }(x_n)\subset\mathbb{R}\\ \text{has a convergent subsequence.}$$ $$\text{Subsequence: }(x_{n_k})\text{ with }n_1 < n_2 < \cdots$$ $$\text{Monotone route: every sequence has a monotone subsequence;}\\ \text{bounded + monotone }\Rightarrow\text{ convergent.}$$ $$\text{Nested route: split a bounded interval repeatedly;}\\ \text{choose a half containing infinitely many terms.}$$
Base points: gray. Subsequence:
blue
line+points. ε-band around L̂:
pink
strip. Nested intervals: faint horizontal bands.