Proof Builder

Line-by-line LaTeX proof workspace

Scratchpad (not saved)

Save to create a persistent, shareable workspace.

Enter adds a newline. Cmd/Ctrl+Shift+Enter adds a new proof line.

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Proof Builder Help

Quick Start

1. Add a theorem title and statement at the top.

2. All inputs are plain text by default.

3. Use `$...$` for inline math and `$$...$$` or `\[...\]` for display math.

4. Proof lines can mix prose, inline math, and display equations in the same field.

5. Use `Enter` for a newline, `Cmd/Ctrl+Shift+Enter` to add and focus a proof line below, and `Tab` / `Shift+Tab` to indent or outdent.

Editing Workflow

1. Drag any line by the row handle area, or use Up/Down buttons.

2. Use Duplicate and Add Below to draft variants quickly.

3. Use the right sidebar snippets for limits, integrals, summations, and symbols.

4. Insert proof templates like Modus Ponens, contradiction, induction, and case split.

Export & Share

1. Copy output as Markdown, plain text, or LaTeX from the toolbar.

2. Use Embed to generate an embeddable version of the current proof.

3. Start a session only when you want collaborative or persisted workflow state.

Example Proofs

Limit of a Convergent Sequence is Unique

Real Analysis

If a sequence converges, it cannot converge to two different limits.

Theorem:

a_{n} \to L

and

a_{n} \to M

, then

L = M

Every Convergent Sequence is Bounded

Real Analysis

A convergent sequence has a bounded tail, and only finitely many early terms remain to control.

Theorem:

a_{n} \to L

, then

(a_{n})

is bounded.

Modus Ponens

Logic

From P and P -> Q, conclude Q.

Theorem:

Q

Irrationality of sqrt(2)

Number Theory

Classic contradiction proof.

Theorem:

2 \in / Q

Limit of x^2 at 2

Real Analysis

Epsilon-delta proof that lim_{x->2} x^2 = 4.

Theorem:

lim_{x \to 2} x^{2} = 4

Kernel is a Subspace

Linear Algebra

Show ker(T) is a subspace of V.

Theorem:

ker (T) \leq V

Eigenvectors for Distinct Eigenvalues are Independent

Linear Algebra

A standard induction proof showing that distinct eigenvalues force independent eigenvectors.

Theorem:

v_{1}, \dots, v_{k}

are eigenvectors of

T

with distinct eigenvalues

λ_{1}, \dots, λ_{k}

, then

v_{1}, \dots, v_{k}

are linearly independent.

Sum of First n Integers

Combinatorics

Induction proof for 1+...+n = n(n+1)/2.

Theorem:

\sum_{i = 1}^{n} i = \frac{n ( n + 1 )}{2}

De Morgan's Law

Set Theory

Show (A \cup B)^c = A^c \cap B^c by double inclusion.

Theorem:

(A \cup B)^{c} = A^{c} \cap B^{c}

Triangle Inequality in R

Metric Spaces

Derive |a+b| \le |a|+|b| from squaring.

Theorem:

∣ a + b ∣ \leq ∣ a ∣ + ∣ b ∣

Uniqueness of Identity

Abstract Algebra

If a group has identity elements e and e', then e=e'.

Theorem:

e = e^{'}

Derivative of a Constant is Zero

Calculus

The difference quotient collapses immediately because the numerator is identically zero.

Theorem:

f (x) = c

for a constant

c

, then

f^{'} (x) = 0

Power Rule for Positive Integers

Calculus

Using first principles, the binomial theorem shows that only the linear-in-h term survives in the derivative limit.

Theorem:

n

is a positive integer and

f (x) = x^{n}

, then

f^{'} (x) = n x^{n - 1}

Chain Rule

Calculus

The derivative of f(g(x)) is f'(g(x)) times g'(x), proved via an extended difference quotient.

Theorem:

(f \circ g)^{'} (x) = f^{'} (g (x)) \cdot g^{'} (x)

Differentiable Implies Continuous

Calculus

Differentiability gives a factorization that forces the function increment to go to zero.

Theorem:

f

is differentiable at

a

, then

f

is continuous at

a

Learn Math

Proofs to Work Through Here

These learn pages pair well with Proof Builder because each one depends on line-by-line structure, explicit assumptions, and clear justification.

Proof that the derivative of a constant function is zero
This proof is short, but it matters because it shows the derivative definition already knows that a function with no change has zero slope. Nothing extra has to be added as a rule.
Proof that every convergent sequence is bounded
This theorem is a standard follow-up to the definition of convergence because it shows how one tail estimate plus finitely many early terms gives a global bound on the whole sequence.
Proof that a differentiable function is continuous
This proof shows why differentiability is a stronger condition than continuity. The difference $f(a+h)-f(a)$ factors into a difference quotient times $h$, and that product is forced to zero.
Epsilon-delta proof that $lim_{x \to 2} x^{2} = 4$
This proof is common because it shows the standard move in early analysis: factor the expression, then bound the extra factor by forcing $x$ into a smaller interval first.
Proof that $\frac{d}{d x} x^{n} = n x^{n - 1}$ for positive integers
This proof shows where the power rule comes from rather than treating it as a black box. The binomial theorem isolates the only term that survives after dividing by $h$ and taking the limit.