How do i prove that the LHS=RHS? i am really confused because of factorials together with pronumerals
I assume that you know
$\displaystyle (k+2)! = (k+1)! \cdot (k+2)$
The common denominator of the fractions therefore is (k+2)!
$\displaystyle 1-\frac1{(k+1)!} + \frac{k+1}{(k+2)!} = 1-\frac{k+2}{(k+1)! \cdot (k+2)} + \frac{k+1}{(k+2)!}$ $\displaystyle \ =\ $ $\displaystyle 1-\left(\frac{k+2}{(k+1)! \cdot (k+2)} - \frac{k+1}{(k+2)!}\right) = 1-\frac{(k+2)-(k+1)}{(k+2)!}$
which will yield the RHS.