A note on market completeness with American put options

We consider a non necessarily complete financial market with one bond and one risky asset, whose price process is modelled by a suitably integrable, strictly positive, càdlàg process $S$ over $[0, T]$. Every option price is defined as the conditional expectation under a given equivalent (true) martingale measure $\mathbb P$, the same for all options. We show that every positive contingent claim on $S$ can be approximately replicated (in $L^2$-sense) by investing dynamically in the underlying and statically in all American put options (of every strike price $k$ and with the same maturity $T$). We also provide a counter-example to static hedging with European call options of all strike prices and all maturities $t\leq T$.