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Example Proofs

Limit of a Convergent Sequence is Unique

Real Analysis

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

Theorem:
If and , then .

Every Convergent Sequence is Bounded

Real Analysis

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

Theorem:
If , then is bounded.

Modus Ponens

Logic

From P and P -> Q, conclude Q.

Theorem:

Irrationality of sqrt(2)

Number Theory

Classic contradiction proof.

Theorem:

Limit of x^2 at 2

Real Analysis

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

Theorem:

Kernel is a Subspace

Linear Algebra

Show ker(T) is a subspace of V.

Theorem:

Eigenvectors for Distinct Eigenvalues are Independent

Linear Algebra

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

Theorem:
If are eigenvectors of with distinct eigenvalues , then are linearly independent.

Sum of First n Integers

Combinatorics

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

Theorem:

De Morgan's Law

Set Theory

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

Theorem:

Triangle Inequality in R

Metric Spaces

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

Theorem:

Uniqueness of Identity

Abstract Algebra

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

Theorem:

Derivative of a Constant is Zero

Calculus

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

Theorem:
If for a constant , then .

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:
If is a positive integer and , then .

Chain Rule

Calculus

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

Theorem:

Differentiable Implies Continuous

Calculus

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

Theorem:
If is differentiable at , then is continuous at .