A common pitfall is failing to show the base case or not clearly stating the inductive hypothesis. Always verify the statement for the initial value and explicitly state your assumption for n=k before proving it for n=k+1.
Be cautious when multiplying or dividing inequalities by negative numbers; remember to reverse the inequality sign. Also, clearly state any assumptions about the signs of variables involved.
A common error is manipulating both sides of the equation simultaneously. Work from one side only, transforming it to match the other side, to maintain logical validity.
To disprove a statement, provide a counterexample. A single instance where the statement doesnt hold true is sufficient to prove it false.
A common mistake is to assume uniqueness from the start. Instead, suppose there are two solutions and show that they must be equal, thus proving uniqueness.
A common error is overlooking the direction of vectors. Ensure that you correctly represent the direction and magnitude when performing vector operations or proving geometric relationships.
Be careful when taking square roots of negative numbers, and always remember to express complex numbers in the correct form (a + bi). Also, pay attention to the argument and modulus when performing operations.
Forgetting to check for differentiability or continuity before applying theorems like the Mean Value Theorem or LHôpitals Rule. Always verify the necessary conditions before applying the theorem.
Clearly define all variables, state any assumptions, and provide justifications for each step. Avoid skipping steps, and ensure your logic is easy to follow.