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In a book that our course is following this semester, the theorem given is only in one direction: if the scheme is both consistent and stable, then the scheme is convergent.
However, since this isn't a double implication theorem, then are there convergent schemes that are either consistent but not stable, or stable but not consistent?
What you're thinking about is the Lax Equivalence Theorem. It is a double implication (if and only if). You can prove generally using a Functional Analysis approach, though that might be more in depth and "further from practical methods" than an approach most classes would take, which is probably why only the forward implication was proved.
In general, I agree with the previous answer by Chris Rackauckas. However, to be more precise, the answer depends on the definition of stability and convergence.
In Stetter "Analysis of discretization methods for ordinary differential equations", in chapter 1, there are simple examples for schemes which are convergent, consistent but not stable or stable but not convergent. They are in a sense degenerate because a bad choice for the spaces is made.
h, in which case Euler is stable (and thus the Lax Equivalence Theorem does show convergence forhsufficiently small). $\endgroup$