Explanation

(d)
We have, $\left| \lambda_{1}a_{1} + \lambda_{2}a_{2} + ... + \lambda_{n}a_{n} \right|$
$\leq \left| \lambda_{1}a_{1} \right| + \left| \lambda_{2}a_{2} \right| + ... + \left| \lambda_{n}a_{n} \right|$
$= \left| \lambda_{1} \right|\left| a_{1} \right| + ... + \left| \lambda_{n} \right|\left| a_{n} \right|$
$= \ \lambda_{1}\left| a_{1} \right| + ... + \lambda_{n}\left| a_{n} \right|\ \ \ (\because each\ \lambda_{k} \geq 0)$
$< \lambda_{1} + ... + \lambda_{n}$
$(\because|a_{k}| < 1\ and\ so\ \lambda_{k}|a_{k}| < \lambda_{k}\ for\ all\ k = 1,\ 2,\ldots n)$
Hence, $|\lambda_{1}a_{1} + \lambda_{2}a_{2} + ... + \lambda_{n}a_{n}| < 1$