Why do we neec to use the triangle inequality and why does this solution imply a contradiction...?I don't understand?

Problem 5. Using the formal definition of limit, prove that lim sin does, not exist. Solution. Suppose for contradiction that lim sin for some L E R. Then, in particular, there exists some o > 0 such that if r E (0, 8), then | sin(1/x) - L|