Argument against exercising an American call early

Explain why it never pays to exercise an American call early when the underlying stock pays no dividends.

Keeping the option is always worth at least as much as exercising. Before expiry the call is worth at least $S-Ke^{-r\tau}\ge S-K$ (with $r\ge0$), so selling it, or holding it and shorting the stock, beats exercise. Early exercise also pays the strike sooner (losing interest) and throws away the protection of not having to buy if the price falls.