# Thread: Convergence of Eulers method.. Differential equations

1. ## Convergence of Eulers method.. Differential equations

Prove that the approximation of Euler's method converges to the exact solution at any fixed point as h->0.
- Because h->0 we know that the number of intervals h approaches infinity. I just can't seem to come up with a way of proving this theorem using algebra.

2. Originally Posted by mathlete2
Prove that the approximation of Euler's method converges to the exact solution at any fixed point as h->0.
- Because h->0 we know that the number of intervals h approaches infinity. I just can't seem to come up with a way of proving this theorem using algebra.
Look at the error analysis, the error per step should be bounded above by $\displaystyle kh^2$, for some $\displaystyle k$ (depending on the function and the interval over which we are integrating) and the number of steps will be $\displaystyle a/h$ where $\displaystyle a$ is the length of the interval we are integrating over. So the total error at a fixed point will be bounded by $\displaystyle kah$, and so will go to zero as $\displaystyle h$ goes to zero.

RonL