Find the limit of the following sequence:

$$
u_{n}=\sum_{i=1}^{n} \frac{1}{F_{i} F_{i+2}}
$$
We find that $\frac{1}{F_{k} F_{k+2}}=\frac{1}{F_{k} F_{k+1}}-\frac{1}{F_{k+1} F_{k+2}}$. Therefore,

$$
u_{n}=\sum_{i=0}^{n} \frac{1}{F_{i} F_{i+2}}=\sum_{i=0}^{n}\left(\frac{1}{F_{i} F_{i+1}}-\frac{1}{F_{i+1} F_{i+2}}\right)=1-\frac{1}{F_{n+1} F_{n+2}}
$$

by telescoping sum.

Since $F_{k}$ tends to infinity, the limit of $u_{n}$ is 1.