Monotonicity of a call price in maturity

How does the value of a call on a non-dividend-paying stock change with its time to expiry? Prove your claim.

It is non-decreasing in maturity (with $r\ge0$). For $T_2>T_1$, at $T_1$ the longer call is worth at least $S_{T_1}-Ke^{-r(T_2-T_1)}\ge S_{T_1}-K$ and at least $0$, so it dominates the payoff of the shorter call; no-arbitrage then gives $C(T_2)\ge C(T_1)$ today. Equivalently, an American call with more time has more rights, and for non-dividend stocks it equals the European.