# Arbitrage with two puts and definition of convexity

This is concerning a common interview style question which has me confused; it has been discussed here: How to Take Advantage of Arbitrage Opportunity of Two Options and Arbitrage opportunity interview question

The question reads "A European put option on a non-dividend paying stock with strike price 80 dollars is currently priced at 8 dollars and a put option on the same stock with strike price 90 dollars is priced at 9 dollars. Is there an arbitrage opportunity existing in these two options?"

The answers indicate that they know an arbitrage opportunity exists because they expect $$Put(λK)<λPut(K)$$, while in this case we have $$Put(80) = (8/9)Put(90)$$. However, I am not sure why we expect $$Put(λK)<λPut(K)$$. I believe the definition of convexity is that $$Put(\lambda K_1 + (1-\lambda) K_2) \leq \lambda Put(K_1) + (1-\lambda) Put(K_2)$$, and plugging in $$K_2=0$$ gives $$Put(λK)\leq λPut(K)$$. Since in the above problem we have $$Put(80) = (8/9)Put(90)$$, convexity is not violated, so what is indicating to us that there is an arbitrage opportunity?