We were shown a video in our class about time dilation which explained a Einstein's theory - the one where he was travelling in a tram and realized that if he travels at light speed, time will flow differently for him. He hypothesized this as the movement of the hands of the clock seemed to get slower with increase in the trams speed. He thought that if he is able to travel at light speed, the hands of the clock will freeze for him.
But, on this, I thought that as Einstein would be travelling at the speed of light, the light reflected from the clock will never reach his eyes. The light reflected from the clock before he achieves light speed, will be the last he sees of the clock's original state, hence, to him, the hands of the clock will appear still, while actually they still will be moving. Time still will be flowing, irrespective of the hands' apparent positions, and at the same rate; only Einstein would not be able to view the indicator(i.e. clock hands) showing this!
Could someone please explain why the theories above are right/wrong?