ISL 1988 Problem 35

A sequence of numbers $a_n, n = 1,2, \ldots,$ is defined as follows: $a_1 = \frac{1}{2}$ and for each $n \geq 2$ $$a_n = \frac{2 n - 3}{2 n} a_{n-1}.$$ Prove that $\sum^n_{k=1} a_k < 1$ for all $n \geq 1.$