Can one stick with explicit ode solver forever? Given enough small time step. Question
Say you have a nonlinear system
$$y' = f(y)$$
where $y \in \mathbb{R}$. You want to integrate it to find $y(t)$. 
Now if you can use arbitrary small time step, can one still use an explicit scheme such as Runge-Kutta-4 to get a stable result? 
Or one must look for implicit schemes?
 A: In theory (exact arithmetic) yes. For instance with the explicit Euler method, if the solution $y$ has a bounded second derivative and $f$ is Lipschitz continuous in its second argument, then the global truncation error is bounded by:
$$
|y(t_n)-y_n|\le {\frac {hM}{2L}}(e^{L(t-t_{0})}-1)\qquad \qquad 
$$
where $M$  is second derivative upper bound and $L$￼ is the Lipschitz constant of $f(t,.)$￼. We see that the error can be made as small as we want by decreasing the step size $h$ (see wikipedia Euler method).
However in practice with a computer using floating number arithmetic (finite precision) you will accumulate rounding errors making your approximate solution meaningless. 
Hence in pratice an implicit method is mandatory for stiff problems (or equivalently as soon as you need a "too small" step size with respect to your floating number precision). 
Also note that in practice there is also another problem: as $\epsilon\rightarrow 0$ you also increase the required number of steps to reach $t>t_0$, thus you increase more and more your computations time.
