Show that zero sequences satisfy the following equation I am working on the following problem and got puzzled: $\\\$ Show that every zero sequence $(a_n), a_n \neq 0$ satisfies the equation: $$ \lim_{n \rightarrow \infty} \frac{\sqrt {1 + a_n} - 1}{a_n} = \frac{1}{2}$$ $\\\$ **My Attempt:** What gets me puzzled is that statement $(a_n), a_n \neq 0$. This basically means a zero sequence does no get the value of zero and as long as it does not get its zero value it satisfies the equation, right? So henceforth we can say this equation should be true for converging sequences? unfortunately I have no clue how to progress with this problem. Thanks for your advice and tipps

$$\frac{\sqrt {1 + a_n} - 1}{a_n} =\frac{\sqrt {1 + a_n} - 1}{a_n}\cdot\frac{\sqrt {1 + a_n} +1}{\sqrt {1 + a_n} +1} =\frac1{\sqrt {1 + a_n} + 1}$$