An American call on a continuous dividend paying stock must be above its intrinsic value, i.e $c(t)\geq\max(S_t-K,0)$.
Why is there a critical price above which it is optimal to exercise (i.e. we have equality in the inequality above)? This is shown in Figure 5.1 in these notes but it is not really explained.
In other words, how can we rule out the situation where $c(t)>\max(S_t-K,0)$ for all prices $S_t$, so no critical price $S_t^*$ exists and the curves in that figure asymptote but never meet?