$1.$ Consider a family of European call options on a non$-$dividend$-$paying stock, with maturity T$,$ each option being identical except for its strike price. The current value of the call with strike price K is denoted by C$($K$).$ There is a risk$-$free asset with interest rate r $>=0.$

$($b$)$ If you observe that the prices of the two options C$($K$1)$ and C$($K$2)$ satisfy K$2$ K$1<$ C $($K$1)$ C $($K$2),$ construct a zero$-$cost strategy that corresponds to an arbitrage opportunity, and explain why this strategy leads to arbitrage.