Convergence of implicit Euler method

[squenshl]

Homework Statement

The implicit Euler method is y_n = y_n-1 + hf(x_n,y_n).
Find the local truncation error and hence show that the method is convergent.

Homework Equations

The Attempt at a Solution

I found the error to be l_n = (-h²/2)y''(x_n-1) + O(h³).
For convergence I am up to using the Lipschitz condition, triangle inequality and ||l_n|| = -Mh²/2 to get:
||e_n|| <= 1/(1-hL)||e_n-1|| - Mh²/(2(1-hL)) for hL <= 1/2 but I am stuck after this. Someone help please.

 

[squenshl]

Since e_o = y(x₀) - y₀ = 0
||e_n|| <= -Mh²/(2(1-hL))(1+(1-hL)+...+(1-hL)^n-1) = -Mh²/(2(1-hL))*((1-hL)^n-1 - 1)/hL