设命题2当n=时成立,即(x)在x0,x+h上连续且 (x)-yb≤b 当n=k+时1(x)=0+/(m(5)d5 由(x,y)在R上连续性知, f(x,94(x)[x02x0+h上连续 从而(x)在x02x+h上连续且 1()y=(0() ≤Mx-x≤M≤b 即命题2当n=k+时成立,从而命题2对所有n都成立设命题2当n = k时成立,即k (x)在[x0 , x0 + h]上连续且 k (x) − y0  b 当n = k +1时  x y f  k  d x x k ( ) ( , ( )) 0 1 0  + = + 由f (x, y)在R上连续性知, f (x,k (x))在[x0 , x0 + h]上连续 从而k+1 (x)在[x0 , x0 + h]上连续且 k+1 (x) − y0 = f  k  d x x ( , ( )) 0  f    d x x   k 0 ( , ( )) 0  M x − x  Mh  b 即命题2当n = k +1时成立, 从而命题2对所有n都成立