A new upper bound on a call option on a dividend paying stock?

 In Options, Futures and Other Derivatives, Chapter 10, Hull derives lower bounds for option prices for European put and call options for dividend paying stocks.

The result in Hull:

For a call option:

$c_0 \geq S_0 - K e^{-rT}$

​For a put option:

$p_0 \geq K e^{-rT} - S_0 $

And he also derives upper bounds for European put and call options for non-dividend paying stocks.

For a call option:

$c_0 \leq S_0$

​For a put option:

$p_0 \leq K e^{-rT}$

​But he doesn't derive upper bounds for European put and call options for dividend paying stocks. And for call options this gives us a tighter upper bound. Also I wasn't able to find this bound online for some reason, it's almost like everyone is just copying these results from other people and not actually deriving them themselves.

The new Result

Let $S_T$ = price of the stock at time $T$.

$K$ = Strike price of the option

$T$ = Maturity of the option

$q$ = dividend yield of the stock

$c_t$ = price of the call option at time $t$.

$max(S_T,K) \leq S_T + K$

$max(S_T,K) - K \leq S_T$

$max(S_T-K,0) \leq S_T$

$c_T \leq S_T$

Then by a no arbitrage argument:

$c_0 \leq S_0 e^{-qT}$

Which is a tighter upper bound than for a non-dividend paying stock.