Put Options and Arbitrage Proof I am trying to prove that if $P$ is the price of a European put option to sell a security whose current price is $S$ with strike price $K$, then $P\le K$ must be true. My first instinct with this is to do a counter proof that shows that if on the other hand $P>K$, then there is an opportunity for arbitrage. How would I go about doing this, and is this the right direction?

This is true assuming the value of the security cannot go negative. If you sell the put, the worst that can happen is that the security will be worthless at expiration, in which case your counterparty will exercise their option to sell it to you at $K,$ which results in a loss of $K$ for you. If you collected more than $K$ in premium, you still win, even in the worst case scenario, so it's an arbitrage opportunity.

(And you get the premium up front and take the losses at expiration, so you also earn interest on the premium.)