For a non-dividend paying European option, the option premium, \(c_t\), is always greater than \(S_t-Ke^{-r\tau}\) where \(\tau=T-t\). This can be shown with some no-arbitrage arguments.

Now for an American option, since it offers everything a European option does and a bit more, the premium at any time should be at least that of the European option, that is \(C_t \geq S_t-Ke^{-r\tau}\). Early exercise of the non-dividend paying option would give a pay off of the intrinsic value \(S_t-K\) but the holder could just sell the option instead for at least \(C_t \geq S_t-Ke^{-r\tau}\) or another way of looking at it is that the option has value of \(C_t \geq S_t-Ke^{-r\tau}\). Since \(C_t \geq S_t-Ke^{-r\tau} >S_t-K\), its value is always greater than the intrinsic value.

So it would always make sense to just hold the option because it has value that is always greater than the intrinsic value.