👉 The proof of mathematical statements, particularly in areas like number theory and algebra, relies on logical deduction using axioms, definitions, and previously proven results. The core involves constructing a sequence of logical steps, starting from known truths (axioms or previously established theorems), to arrive at the conclusion in a way that is both valid and irrefutable. This process ensures the statement's truth is not just assumed but rigorously demonstrated, often employing techniques like direct proof (showing the statement directly follows from the premises), proof by contradiction (assuming the opposite leads to a logical inconsistency), or mathematical induction (proving for base cases and showing a property holds for the next case if it holds for the current one). The elegance of a proof lies in its ability to convey the necessity and universality of the statement, making it a cornerstone of mathematical reasoning and communication.
