My text asks whether the following statement is true or false:
The forward Euler method for approximating the solution of $x'=\lambda x$ is stable for all $\lambda \in \mathbb R$ and all step sizes $0<h\leq \frac{2}{|\lambda|}$.
We know that the forward Euler method is unconditionally stable for $\lambda \geq 0$. And if $\lambda<0$ then it's conditionally stable, with $0<h\leq \frac{2}{|\lambda|}$ being the condition.
So the answer to the question seems to be "true". Yet the professor is known for tricky question, so I guess $\lambda = 0$ would cause problems in the question statement (unless we assume $\frac{2}{0}:=\infty$, which we shouldn't in the simple Intro to Numerics course). Another issue I thought of: the problem for the method should be an initial value problem but $x(0)=x_0$ is nowhere to see.
Or putting the initial value issue aside we can write:
$$\forall \lambda \in \mathbb R \ \ \forall h \in \mathbb R \ \ \left(0<h\leq \frac{2}{|\lambda|} \implies \text{the method is stable} \right)$$
Say, if $\lambda = 0$, what happens with the implication from "not defined"?
Am I over thinking it? Is the statement true/false?