simple yet clever proof with factorials

we're supposed to use the principle of mathematical induction for this.

http://dl.getdropbox.com/u/594924/fac.jpg

basically the inductive part boils down to proving algebraically that

$\displaystyle (k+1)!-1+(k+1)\cdot(k+1)!=(k+2)!-1$

but i'm not familiar enough with factorials to show this. i've read about factorials on wikipedia but they didn't have any good identities or anything i could use. any suggestions would be very helpful but for now i'm going to just stare at it for a few minutes and read about series with factorials.

thanks guys