The diagram shows the curve with equation \(y=\ln x\) for \(x \geqslant 1\), together with a set of \((N-1)\) rectangles of unit width.
(a) By considering the sum of the areas of these rectangles, show that
\[\ln N!>N \ln N-N+1 .\]
(b) Use a similar method to find, in terms of \(N\), an upper bound for \(\ln N!\).