Building a cash digital from vanilla options

Using only vanilla options, how would you replicate a digital call that pays $1$ if the spot ends above $K$?

Use a narrow call spread scaled by its width. Buy $1/\varepsilon$ calls struck at $K-\varepsilon$ (or $K$) and sell $1/\varepsilon$ calls at $K$ (or $K+\varepsilon$). The payoff is $0$ below the lower strike, $1$ above the upper one and linear in between; as $\varepsilon\to0$ it tends to the digital, so its price is $-\partial C/\partial K$. Traders keep $\varepsilon$ finite and choose the side that over-hedges.