Moving from calculation to proof can feel like changing subjects. In an earlier course, an answer might be a number. In linear algebra or real analysis, you may need to explain why a statement holds for every object that meets a set of conditions.
Translate the statement carefully
Write down what is given and what must be shown. Underline words such as ‘every,’ ‘there exists,’ and ‘if.’ They shape the task. A claim about every vector needs an argument that works generally; a claim that one vector exists may need only a suitable construction.
Return to the definitions before looking for a clever trick. If a problem asks whether a mapping is linear, write the conditions involving addition and scalar multiplication. A definition often supplies the structure of the proof more directly than a remembered example.
Try examples without treating them as proof
Small cases can help you see what a statement means. They can also expose a false claim. For instance, testing a transformation on simple vectors may suggest how it behaves. But several successful examples cannot establish a universal statement.
Use examples to generate a plan, then ask what part of that plan holds for an arbitrary object. If the claim is false, a single valid counterexample is enough to disprove it. Make sure your counterexample actually meets the assumptions; otherwise it does not address the statement.
Make the logical links visible
A proof should tell the reader why each important step follows. Phrases such as ‘by the definition of continuity’ or ‘because the set is closed under addition’ are useful when they name the actual reason. Avoid inserting ‘obviously’ where you are uncertain.
When reading a proof, pause before the next line and predict what might be useful. If a step surprises you, identify the definition, result, or rearrangement behind it. This is slower than copying the proof, but it gives you material you can use on a different question.
Use tutoring for the difficult transition
Bring your draft rather than only the final theorem. A tutor can ask why a step is justified, whether an assumption has been used, or whether the argument proves the intended direction. You should still write and own the final reasoning.
Choose a narrow session goal: understanding proof by contradiction, checking a linear independence argument, or unpacking a limit definition. An hour is more productive when the logical problem is visible. Leave with a similar statement to attempt independently, and compare the structure rather than the wording of the two proofs.
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