[tex]\dfrac{1}{50}<\dfrac{1}{10}\leq \dfrac{1}{10} \\ \\ \dfrac{1}{50}<\dfrac{1}{11}<\dfrac{1}{10} \\ \\ \dfrac{1}{50}<\dfrac{1}{12}<\dfrac{1}{10} \\ \\ .........................\\ \\ \dfrac{1}{50}<\dfrac{1}{49}<\dfrac{1}{10} \\ \\\\\text{Adunam inegalitatile:}[/tex]
[tex]\dfrac{1}{50}+\dfrac{1}{50}+...+\dfrac{1}{50}< \dfrac{1}{10}+\dfrac{1}{11}+\dfrac{1}{12}...+\dfrac{1}{49}<\dfrac{1}{10}+\dfrac{1}{10}+...+\dfrac{1}{10}\\ \\ \text{Observam ca de la 10 la 49 sunt 40 de termeni.} \\ \\ 40\cdot \dfrac{1}{50}<\dfrac{1}{10}+\dfrac{1}{11}+\dfrac{1}{12}...+\dfrac{1}{49}<40\cdot \dfrac{1}{10} \\ \\ \dfrac{40}{50}<\dfrac{1}{10}+\dfrac{1}{11}+\dfrac{1}{12}...+\dfrac{1}{49}<\dfrac{40}{10} \\ \\ \\\Rightarrow \dfrac{4}{5}<\dfrac{1}{10}+\dfrac{1}{11}+...+\dfrac{1}{49}<4[/tex]
