The Calendar-Spread-Inequality compares the prices of two European Call Options on the same underlying non-dividend-paying stock, but with different maturities $$T_1. Denote the value of a call option with strike $$K$$ and maturity $$T$$ at time $$t\leq T$$ as $$C_K(t,T)$$ (a put will be denoted by $$P_K(t,T)$$). The calendar-spread-inequality then states:

$$C_{K'}(t,T_1)\leq C_K(t,T_2),$$

where $$K'=Ke^{-r(T_2-T_1)}$$.

To prove this we first consider the Monotonicity Lemma (see "An Introduction to Quantitative Finance" by Stephen Blyth), which states that, under the no-arbitrage assumption, if two portfolios, $$A$$ and $$B$$, have values $$V^A(T')\leq V^B(T')$$ at time $$T'$$, their values must obey $$V^A(t)\leq V^B(t)$$ at any time $$t\leq T'$$. (A proof will not be given here.)

Now, consider two portfolios, $$A$$ and $$B$$, with $$A=\{\text{own one call with strike K' and maturity T_1}\}$$ and $$B=\{\text{own one call with strike K and maturity T_2}\}$$. At time $$T_1$$ we have $$V^A(T_1)=\text{max}\{0,S_{T_1}-K'\}$$.

According to Put-Call-Parity for European options: $$C_K(t,T)-P_K(t,T)=S_t-Ke^{-r(T-t)}$$, which means that \begin{align} C_{K'}(T_1,T_1)=V^A(T_1)&=\text{max}\{0, S_{T_1}-Ke^{-r(T_2-T_1)}\}\\ &=\text{max}\{0,C_K(T_1,T_2)-P_K(T_1,T_2)\}\\ &\leq C_K(T_1,T_2)=V^B(T_1), \end{align} since $$P_K\geq0$$. By the monotonicity lemma we find that $$C_{K'}(t,T_1)\leq C_K(t,T_2)$$ for $$t\leq T_1$$.

Is this proof correct or am I missing something?

Here is a shorter proof with $$r=0$$ which I hope is helpful for you as well. I'll leave $$r\neq0$$ for you to generalise. Let $$T > t$$, then:
\begin{align} E_0 \left[ (S_T - K)_+ \right] &= E_0 \left[ E_t \left[ (S_T - K)_+ \right] \right] \\ &\geq E_0 \left[ ( E_t (S_T) - K)_+ \right] \\ &= E_0 \left[ ( S_t - K)_+ \right]. \end{align}