I have found a proof that an American put option without dividend will never be exercised early. However, I suspect that that is not true, so there should be a mistake in the proof. The proof is as follows:

Consider an American put option $P$ without dividend. Let the strike price be $E$ and let $S$ be the underlying stock. Suppose that the holder of the option would exercise the option, which would yield him $E-S$. Then he could instead sell the option to someone else. Since the option price is always at least $E-S$, this would give him at least as much money. Therefore, we can assume that the option is never exercised early.

Could you help me find the mistake in this proof?

I have found a proof that an American put option without dividend will never be exercised early. However, I suspect that that is not true, so there should be a mistake in the proof. The proof is as follows:

Consider an American put option $P$ without dividend. Let the strike price be $E$ and let $S$ be the underlying stock. First we prove that the price of the option is at least $E-S$. We do this by contradiction, so suppose that the option price is smaller than $E-S$. Then the following arbitrage option would occur: someone could buy the put option and 1 times the stock $S$. Then, the person could immediately exercise the put option. This would give the person an immediate risk-free profit. Therefore this can't be the case, hence the option price is at least $E-S$.

Next, suppose that the holder of the option would exercise the option, which would yield him $E-S$. Then he could instead sell the option to someone else. Since the option price is always at least $E-S$, this would give him at least as much money. Therefore, we can assume that the option is never exercised early.

Could you help me find the mistake in this proof?