= Solution
Under the usual no-<arbitrage> interpretation from part (a), let $Q^N$ be an <equivalent martingale measure> for the strictly positive <numéraire> $N$. Then $M_t=S_t/N_t$ is a <martingale>. Define the discounted <European call option> payoff $Y_t=(S_t-K)^+/N_t=(M_t-K/N_t)^+$. Because $N_{t+1}\geq N_t$ and $K>0$, we have $K/N_{t+1}\leq K/N_t$. The <conditional Jensen inequality> therefore gives
$$
\begin{aligned}
\mathbb E^{Q^N}[Y_{t+1}\mid\mathcal F_t]
&\geq\mathbb E^{Q^N}[(M_{t+1}-K/N_t)^+\mid\mathcal F_t]\\
&\geq\big(\mathbb E^{Q^N}[M_{t+1}\mid\mathcal F_t]-K/N_t\big)^+
=Y_t.
\end{aligned}
$$
For the middle step, the strike $K/N_t$ is fixed conditionally on $\mathcal F_t$; alternatively use $\mathbb E[X^+\mid\mathcal F_t]\geq\max\{\mathbb E[X\mid\mathcal F_t],0\}$. Thus $Y$ is a <submartingale>. <Risk-neutral valuation> in units of $N$ gives $C(T,K)=N_0\mathbb E^{Q^N}Y_T$, with the usual fixed initial prices. \b[Consequently]
$$
\boxed{C(T+1,K)\geq C(T,K).}
$$
The source's word “increasing” means nondecreasing: a <European call option> can have zero payoff at successive maturities, so strict increase is not guaranteed. Integrability is understood in the claim class for which the stated replication prices exist.
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