[tex]\displaystyle Relatia ~\sin x\ \textless \ 0~este~satisfacuta~daca~ \\ \\ x \in \Big( (2k+1) \pi; (2k+2) \pi \Big),~unde~ k \in \mathbb{Z}. \\ \\ Vom~demonstra~ca~orice~astfel~de~interval~contine~cel~putin \\ \\ un~numar~natural~\forall~k \geq 0. \\ \\ Observam~ca~(2k+2) \pi-(2k+1) \pi=(2k+2-2k-1) \pi= \pi, \\ \\ deci~intervalul~are~lungimea~\pi\ \textgreater \ 3,~ceea~ce~inseamna~ca~el \\ \\ contine~cel~putin~un~numar~natural.[/tex]
[tex]\displaystyle Deci~pentru~fiecare~ k \in \mathbb{Z}~exista~cel~putin~un~numar~natural~ \\ \\ inclus~in~intervalul~mentionat~anterior.~Prin~urmare,~ \\ \\ inegalitatea~ \sin n \ \textless \ 0~este~verificata~pentru~ \underline{o~infinitate~de } \\ \\ \underline{numere~naturale}.[/tex]
