$1.$ Let w be an arbitrary string, and let n $=|$w$|.$ Prove each of the following statements.

$($a$)$ w has exactly n $+1$ prefixes.

$($b$)$ w has exactly n proper suffixes.

$($c$)$ w has at most n$($n $+1)/2$ distinct substrings. $($Why $$at most$?)$

$($d$)$ w has at most $2$

n $1$ distinct proper subsequences. $($Why $$at most$?)$