Give a concrete sequence of rationals which converges to an irrational number and vice versa.

$a_n = \left( 1 +\dfrac{1}{n} \right)^n $ converges to $e$

How about $a_n$ = the decimal expansion of $\sqrt{2}$ up to the $n$-th place

A formal definition could be $$a_n = \lfloor 10^n\sqrt{2} \rfloor 10^{-n} $$

Take ratios of consecutive Fibonacci numbers: $\frac11,\frac21,\frac32,\frac53,\frac85,\dots$. It is well known that this converges to the golden ratio $\frac{1+\sqrt{5}}{2}$, which is irrational.