Paying one if spot ends between 100 and 110

A contract pays $1$ at expiry if the spot lies between $100$ and $110$, and nothing otherwise. Show how to construct it from ordinary calls.

Long a digital at $100$, short a digital at $110$, each built as a tight call spread. With a small $\varepsilon$, buy $1/\varepsilon$ calls at $100-\varepsilon$, sell $1/\varepsilon$ at $100$, sell $1/\varepsilon$ at $110-\varepsilon$ and buy $1/\varepsilon$ at $110$. The payoff equals the range digital except on two strips of width $\varepsilon$, so it replicates exactly in the limit; the price is about $-\partial_K C(100)+\partial_K C(110)$.