ODE Euler approximation converging to exact solution

Question

I have this ODE: $dy/dt = -y$, $y(0)=10$. How can I prove that its Euler numerical solution converges to the exact solution?

This is what I have attempted: The Euler's scheme for this would be: $y_n = y_{n-1}(1 - \Delta_t) = y_0(1 - \Delta_t)^n$. As $\Delta_t$ goes to $0$, $y_n$ would go to $y_0$, which is 10.

The exact solution would be $y(t)=10e^{-t} = 10e^{-n\Delta_t}$. As $\Delta_t$ goes to $0$, t goes to $0$ (I don't know if this is valid), and thus the solution converges to 10 as well.

I am really unsure about what I have done. Can someone help me with this? Thank you!

Answer 1

You need to increase $n$ so that $nΔt=T$ remains constant, as you want to compare the numerical appriximations at the same time point. Then you get the more familiar limit $$ \lim_{n\to\infty}\left(1-\frac{T}{n}\right)^n. $$

Answer 2

For the sake of illustration, the plot shows the functions

$$10\left(1-\frac tn\right)^n$$ for $n=1,2,3,4$. The dark green curve is the exponential solution.