I'm just starting to learn about special relativity, and I'm a little bit confused about something. Take the example of an observer in $S$ on the ground observing a train move at constant velocity $v$ relative to $S$, an observer in $S'$ is on the train, and this observer in $S'$ flashes a light that reflects from the ceiling and returns to him in a time, $t'$ he measures.
I know that in general, if two frames $S$ and $S'$ are in relative uniform motion with respect to each other, and an observer in $S$ can see the clock of an observer $S'$ and the observer in $S'$ can see the clock of the observer $S$, then the observer in $S$ will see the clock of the observer in $S'$ run slower than his own, and vice versa. But I also know that $S'$ time is proper time, and so $t'\leq t$. This seems very strange to me. Observer $S$ sees his clock running faster, so intuitively, observer $S$ expects to measure less time for the event, but the time he measures for the light to return is no less than the time that observer $S'$ measures.
Am I understanding this correctly?