Solved and commented exercises

Each solution is written step by step, with the goal of making the reasoning transparent and accessible. Rather than stating results abruptly, the proofs are built progressively, explaining not only what is done, but also why it is done.

The approach is meant to be pedagogical, patient, and structured. In these solutions, you will typically find:

• a careful analysis of the statement and assumptions;
• rigorous proofs, without logical gaps;
• key reminders embedded throughout the reasoning, without assuming everything is known in advance;
• intuitive insights to better understand the underlying ideas;
• alternative methods when they provide complementary perspective;
• and sometimes, commented unsuccessful attempts, to illustrate what does not work and why.

When the source of an exercise is known, I indicate it in order to place the statement back in its original context (book, exam, etc.).

These solutions are intended for anyone who wants to deepen their understanding of mathematical reasoning — whether you are an undergraduate student, in a preparatory program, or simply curious.