Proofs
Browse a catalog of interesting and common proofs, from core analysis and linear algebra arguments to standard proof techniques worth knowing well.
Filter By Subject
Proof that the limit of a convergent sequence is unique
A convergent sequence cannot settle down to two different numbers. The proof is short, but it is a good model for how contradiction and the definition of limit work together.
Open proof →
Proof that the kernel of a linear map is a subspace
This is a standard linear algebra proof because it packages the subspace test into one reusable pattern: show zero is inside, then check closure under addition and scalar multiplication.
Open proof →
Proof that eigenvectors for distinct eigenvalues are linearly independent
Distinct eigenvalues do more than label eigenspaces. They force eigenvectors to be independent, which is exactly why diagonalization becomes possible so often in practice.
Open proof →
Epsilon-delta proof that
This proof is common because it shows the standard move in early analysis: factor the expression, then bound the extra factor by forcing into a smaller interval first.
Open proof →
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.
Open proof →
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.
Open proof →
Proof that 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 and taking the limit.
Open proof →
Proof of the chain rule
The chain rule says the derivative of is . The proof uses a continuous extension of the difference quotient to handle all cases cleanly.
Open proof →
Proof that a differentiable function is continuous
This proof shows why differentiability is a stronger condition than continuity. The difference factors into a difference quotient times , and that product is forced to zero.
Open proof →