# Read induction proofs in Book Isle: align base case, hypothesis and next step
Language: en
Canonical: [Read induction proofs in Book Isle: align base case, hypothesis and next step]()
中文：[中文 HTML]()
English: [English HTML]()
Published: 2026-11-07T00:00:00+08:00
Modified: 2026-11-07T09:41:37+08:00
Author: Bison
Use a positive-integer sum to check scope, where P(k) is used and the k+1 target, separating numerical checks from a general proof.
English translation of the [Chinese original](), translated on 2026-11-07. Original publication dates, attribution, code and version notes are retained.
Assuming a statement in an induction step does not declare every desired conclusion true. Retain theorem and proof locations in Book Isle and annotate base case, conditional hypothesis and next-case derivation in your own notes. Use a classical positive-integer sum to practise reading and checking, not an automatic proof engine or an allegedly new proof. A source may use different notation and starting index; rematch its scope instead of applying a universal answer whenever induction appears.
## Retain the whole statement and integer domain
Choose a short induction proof you can understand and record title, edition, chapter, proposition number, beginning/end phrases and adjacent conditions. State what n is, its starting point and all targeted values, not only the equation. Here n is an integer n≥1 and P(n) is 1+2+…+n=n(n+1)/2; the ellipsis is a finite consecutive sum ending at n. If the source begins at zero or uses another set, retain its definition instead of imposing this start. Keep statement separate from proof so every step targets the same proposition rather than silently changing it.
## Substitute into both sides of the base case
For n=1, the left side is 1 and the right is 1×(1+1)/2=1, establishing P(1). Write the substitution instead of hiding an unclear step under obvious. The base need not always be one; use the source statement’s first allowed value. A sole n=0 base for this example would need an explained connection to n≥1, not an unannounced change. If substitution encounters division by zero or undefined objects, return to the domain rather than treating a calculator message as the condition. This establishes only the starting case; a general induction step remains necessary.
## Treat the hypothesis as a conditional case for arbitrary k
Choose an arbitrary integer k≥1 and temporarily assume P(k): 1+2+…+k=k(k+1)/2. Retain arbitrary, domain and assumption instead of writing that all n are already known. The hypothesis establishes a conditional passage between cases, not a completed conclusion. Fixing k=3 cannot prove the general step; numbers only help check arithmetic. Map a source’s m or other letter explicitly rather than switching without explanation. Mark the sum the hypothesis may replace and its scope, then identify its actual use next.
## Write the complete next-case target
Write P(k+1) separately: 1+2+…+k+(k+1)=(k+1)(k+2)/2. Substituting k+1 for n in the whole formula makes the second factor k+2, not k+1. State the target without treating it as a known equality. The left adds one term k+1, not merely 1 to the sum and not 1 to every summand. Verify unclear notation on the page or explain in words rather than dropping parentheses from memory. Mark this target still to derive so the next line cannot circularly use it as its own justification.
## Identify the exact substitution of the hypothesis
Start with the next left side, replace its first k terms using the hypothesis k(k+1)/2, then add (k+1). A common denominator gives \[k(k+1)+2(k+1)\]/2; factoring gives (k+1)(k+2)/2, the target. Label each equality substitution, common denominator or factoring instead of copying results alone. If 2(k+1) loses its 2, inspect the mishandled denominator; if the result is (k+1)²/2, inspect k+2. Ordinary algebra follows the one hypothesis substitution; do not use the unproved next case backwards to prove itself.
## Conclude from both base and general step
With P(1) true and P(k) implying P(k+1) for arbitrary k≥1, induction establishes all integers n≥1. Checking 1, 2 and 3 cannot replace the general step, nor can the implication replace the base. For a simple counterexample, the target all n≥1 satisfy n≥2 has a valid preservation implication to the next integer but a false n=1 base, so it is not proved. This illustrates the need for a base without accusing another textbook. Retain the domain rather than extending to real numbers or arbitrary starting points beyond this proof.
## Do a numerical check and a source check separately
Check n=3: 1+2+3=6 and 3×4/2=6; then n=4: 10 and 4×5/2=10. These catch transcription/arithmetic errors rather than supply a universal proof. Leave the reader and return via bookmarks or phrases to statement, base and induction step, checking arbitrary, assume, implies and domain. Reopen exported notes and inspect parentheses, additions, denominator and k+2; full plain-language formulas may substitute for difficult entry. Rematch content after changed proposition numbers instead of old pages alone. Distinguish interpretation from source excerpts so later corrections identify understanding versus copying.
## Keep a proof card that explains the connections
Retain domain, P(1), P(k) hypothesis, P(k+1) target, hypothesis-use line and conclusion. With the source hidden, explain why the added term is k+1, why k+2 appears and why three numerical cases are insufficient. This exceeds memorising a neat formula. Keep an unclear equality open rather than compressing it into obvious. Recheck start and statement for each new proof instead of copying every line here. Book Isle locates and stores the reading; the card records one understood argument, not automatic validation of all proofs or replacement for mathematical study.
## Related tutorials and sources
[Follow worked examples in Book Isle: track symbols, units and substitutions](); [Book Isle definition reading: necessary conditions, sufficient conditions and counterexamples]().
Sources checked on 7 November 2026: the [official Book Isle App Store listing]() and [Apple Lookup]() show version 1.1.4, requiring iOS / iPadOS 15.6 or later. This workflow uses local reading, search, bookmarks and notes for manual checking. Readers record the samples and conclusions; automatic chart recognition, statistical testing or proof verification is not assumed. Confirm available controls against the installed version and file type. The exercise does not claim step-by-step device testing in this batch.
Method reference: [Oxford Mathematical Institute: natural numbers and induction](). The sum is classical; Book Isle locations, line labels and self-check records are an original reading exercise, not an official solution from that course.