= Solution
The <Black-Scholes implied volatility> of a traded call price $C$ is the volatility $\sigma$ which makes the <Black-Scholes formula> equal to $C$. For
$$
(S_0-Ke^{-rT})^+<C<S_0,
$$
the price is a continuous strictly increasing function of $\sigma>0$, by its positive <vega>, with precisely these endpoint limits. The <intermediate value theorem> and strict monotonicity therefore give a unique positive implied volatility. The lower endpoint corresponds to the limiting value $\sigma=0$; the upper endpoint is reached only as $\sigma\to\infty$. Prices outside the bounds admit the <arbitrages> above and have no implied volatility.
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