Answer
$\{a_{n}\}$ converges to $1$.
Work Step by Step
Let $\{b_{n}\}=\{1,1,1,...\}.\qquad \{b_{n}\}\rightarrow 1$
Let $\displaystyle \{c_{n}\}=\{\frac{1}{n}\}.\qquad \{c_{n}\}\rightarrow 0$
$\{b_{n}\}$ and $\{c_{n}\}$ are both convergent, and
$a_{n}=b_{n}-c_{n}$.
By Theorem 1.2, $\{a_{n}\}$ converges to $1-0=1$.
