Convergent sequence of irrational numbers that has a rational limit. Is it possible to have a convergent sequence whose terms are all irrational but whose limit is rational? 

If $q$ is any rational number at all and $n$ is a positive integer then $q+\frac 1 n \sqrt 2$ is irrational (it's a simple algebra exercise to prove that), and $\lim\limits_{n\to\infty}\left(q + \frac 1 n \sqrt 2\right) = q$.
