Proof to part b:
Base: Where n=1:
LHS=1/2!
RHS=((1+1)!-1)/(1+1)!=(2!-1)/2!=1/2!=LHS
Now we assume the statement holds true for n=k, and show that it holds true for n=k+1
So we have
1/2!+2/3!+3/4!+4/5!+⋯+k/(k+1)!=((k+1)!-1)/(k+1)!
And want to show that
1/2!+2/3!+3/4!+4/5!+⋯+k/(k+1)!+((k+1))/(k+2)!=((k+2)!-1)/(k+2)!
LHS=1/2!+2/3!+3/4!+4/5!+⋯+k/(k+1)!+((k+1))/(k+2)!
=((k+1)!-1)/(k+1)!+((k+1))/(k+2)!
=((k+2)!(k+1)!-(k+2)!+(k+1)(k+1)!)/(k+1)!(k+2)!
=((k+2)!(k+1)!-(k+2)(k+1)!+(k+1)(k+1)!)/(k+1)!(k+2)!
=((k+2)!-(k+2)+(k+1))/(k+2)!
=((k+2)!-k+k-2+1)/(k+2)!
=((k+2)!-1)/(k+2)!
Q.E.D.