Define \begin{align} c & \text{ the price of a European call option at time } t = 0, \\ S & \text{ the spot price of a share at time } t=0, \\ X & \text{ the strike price for the option}, \\ T & \text{ the expiry time of the option}, \\ r & \text{ the } T \text{-year spot interest rate}, \\ S_T & \text{ the spot price of the share at time } t=T. \end{align} My course notes contain a proposition,
Assume that the stock does not pay dividends. Then $c>S-Xe^{-rT}$.
To prove this consider:
Portfolio A: one European call option plus an amount of cash equal to $Xe^{−rT}$ deposited for $T$ years at an interest rate of $r$.
Portfolio B: one share.
After $T$ years portfolio A will yield an amount of cash equal to $X$. If, after $T$ years, the stock price $S_T$ is above $X$, the call option in portfolio A will be exercised, the share sold and the portfolio will be worth $S_T$ . Otherwise, after $T$ years, $S_T ≤ X$, the option is not exercised and the portfolio will be worth $X$. So after $T$ years portfolio A is worth max$(S_T, X) \geq S_T$, and since portfolio B is always worth $S_T$ after $T$ years, the initial value of portfolio A must be no less that the initial value of portfolio B, which is just $S$. But since sometimes portfolio A is worth more than portfolio B we have a strict inequality $c+Xe^{−rT}>S$.
In cases where $S_T>X$ we have $V(A)=V(B)$ at time $t=T$, which means in those cases we have $V(A)=V(B) \; \forall \; t \in[0,T]$; so I assume "sometimes portfolio A is worth more than portfolio B" doesn't mean "in every case there is a time at which portfolio A is worth more than portfolio B". The other interpretation I can think of is "there are cases in which at time $t=T$ portfolio A is worth more than portfolio B". If that's the intended meaning, why is the inequality made strict without specifying that restriction?
The correct way of interpreting the last statement is by adding probabilities (to an otherwise model-free argument). Let $V_T^A$ and $V_T^B$ be the terminal values of portfolios $A$ and $B$, respectively. Then, the conclusion of your argument can be translated to: $V_T^A \geq V_T^B$ and $P(V_T^A > V_T^B) > 0$.
Thus, the initial value of portfolio $A$ must be strictly larger.