What is wrong with the Lax-Richtmyer fundamental theorem of linear numerical analysis ?

We show that the celebrated 1956 Lax-Richtmyer linear theorem in Numerical Analysis - often called the Fundamental Theorem of Numerical Analysis - is in fact wrong. Here "wrong" does not mean that its statement is false mathematically, but that it has a limited practical relevance as it misrepresents what actually goes on in the numerical analysis of partial differential equations. Namely, the assumptions used in that theorem are excessive to the extent of being unrealistic from practical point of view. The two facts which the mentioned theorem gets wrong from practical point of view are : - the relationship between the convergence and stability of numerical methods for linear PDEs, - the effect of the propagation of round-off errors in such numerical methods. The mentioned theorem leads to a result for PDEs which is unrealistically better than the well known best possible similar result in the numerical analysis of ODEs. Strangely enough, this fact seems not to be known well enough in the literature. Once one becomes aware of the above, new avenues of both practical and theoretical interest can open up in the numerical analysis of PDEs.