We consider two put options P$1($S$,$ t$)$ and P$2($S$,$ t$)$ with exercise prices E$1$and E$2,$ respectively,where E$1<$ E$2,$ and with the same expiry date T$.$ The interest rate is fixed at r $>0.($a$)$ Show by a no$-$arbitrage argument that P$1($S$,$ t$)<=$P$2($S$,$ t$)$ for all S $>0$ and t$<=$T$.($b$)$ Show by a no$-$arbitrage argument that P$2($S$,$ t$)$P$1($S$,$ t$)<=$e$^$r$($T$$t$)($E$2$E$1)$ for all S $>0$ and t$<=$T$.$