Explain briefly what is meant by the convergence of a numerical method for solving the ordinary differential equation

$$y^{\prime}(t)=f(t, y), \quad t \in[0, T], \quad y(0)=y_{0} .$$

Prove from first principles that if the function $f$ is sufficiently smooth and satisfies the Lipschitz condition

$$|f(t, x)-f(t, y)| \leqslant L|x-y|, \quad x, y \in \mathbb{R}, \quad t \in[0, T],$$

for some $L>0$, then the backward Euler method

$$y_{n+1}=y_{n}+h f\left(t_{n+1}, y_{n+1}\right)$$

converges and find the order of convergence.

Find the linear stability domain of the backward Euler method.