A-stability
Some methods stay bounded only if you keep the step small; others stay bounded no matter how big a step you take, as long as the true solution is decaying. That second, gold-standard property is A-stability: a method is A-stable if it never blows up on any decaying problem, for ANY step size. It is the quality you demand of a method meant for stiff problems.
Precisely, recall the stability region: the set of z = lambda*h for which the method's amplification |R(z)| <= 1. A method is A-STABLE if this region contains the ENTIRE left half of the complex plane — every z with negative real part. Since the test equation y' = lambda*y decays exactly when lambda has negative real part, A-stability says: whenever the true solution decays, the numerical one does too, regardless of h. Backward Euler is A-stable (its region is the whole left half-plane and beyond); the trapezoidal rule (Crank-Nicolson) is A-stable and sits right on the boundary; forward Euler and all explicit Runge-Kutta methods are NOT (their regions are small bounded blobs). Among multistep methods, only BDF1 and BDF2 are A-stable.
A-stability is what frees you from the tyranny of stiffness: with an A-stable method you choose the step for ACCURACY alone, never for fear of explosion, even when the equation has wildly fast decaying modes. But be honest about what it does and does not give. First, A-stability is only a condition for not blowing up — it says nothing about accuracy, and an A-stable method with a big step can return a stable but very wrong answer. Second, Dahlquist's second barrier proves no linear multistep method of order above 2 can be A-stable, which is why high-order stiff solvers use the slightly weaker A(alpha)-stability or implicit Runge-Kutta methods. Third, A-stability can be TOO strong a damper for oscillatory problems (an even stricter notion, L-stability, is wanted for the very stiffest decay, while symplectic methods are wanted for conservative oscillation).
Take y' = -1000y, true solution decaying fast to 0. With backward Euler and a comfortable h = 0.5: y_{n+1} = y_n/(1 + 500), each step shrinks by 1/501, bounded and decaying — A-stability in action. Forward Euler at the same h = 0.5 gives factor 1 - 500 = -499, so |R| = 499 and the solution explodes; it would need h < 0.002 to stay stable.
A-stable: bounded for every decaying mode at any step size — the property stiff solvers need.
A-stability guards against blow-up, not against inaccuracy: an A-stable method with a large step can be perfectly bounded yet far from the true solution. Dahlquist's second barrier also limits A-stable multistep methods to order 2, a fundamental ceiling, not a missing cleverness.