Convergence of implicit Euler method

squenshl
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Homework Statement


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


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The Attempt at a Solution


I found the error to be ln = (-h2/2)y''(xn-1) + O(h3).
For convergence I am up to using the Lipschitz condition, triangle inequality and ||ln|| = -Mh2/2 to get:
||en|| <= 1/(1-hL)||en-1|| - Mh2/(2(1-hL)) for hL <= 1/2 but I am stuck after this. Someone help please.
 
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Since eo = y(x0) - y0 = 0
||en|| <= -Mh2/(2(1-hL))(1+(1-hL)+...+(1-hL)n-1) = -Mh2/(2(1-hL))*((1-hL)n-1 - 1)/hL
 
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