This sort of question has been asked a few times here since I arrived.
Other answers above say pretty much the same thing, but this is how it was explained to me during Mathematics lessons at university.
Firstly you have to understand that there are essentially two types of infinity :
Countable (1 to infinity) and Uncountable (0.1, 0.01, 0.001... infinity)
Between any two moments, there is an uncountable infinity of moments.
Each second can be broken down into tenths, hundredths, thousandths.. etc.
Infinity is something we live in, on a daily basis.
This has confused people for centuries, not least mathematicians and physicists.
Others mention the very first attempt to grapple with infinity, which we know formally as
zeno's paradox (achilles and the tortois)
The real problem with that paradox is that the premise of the paradox is wrong.
That is, it is defined in terms of 'distance' and not 'velocity'.
Eventually this paradox was solved using a mathematical technique called Sums of Series, or more formally by a whole branch of mathematics that deals with 'limits' and infinitesimals which we know as 'calculus' (see the 'significance' section of this article).
Infinity is at it's very essence a human concept.
Infinity in nature is never really observed. There are no 'naked singularities'.
I'm talking about Sir Roger Penrose a lot recently, but this is a statement of his.
In fact the universe seems to go to great lengths to hide singularities from us.
There exists a thing called the 'planck length'.
The smaller the thing you wish to inspect, the more energy you are required to apply during inspection. Eventually, at very small distances (the planck length) the energy required to observe that distance becomes so great that spacetime itself collapses and hides away from inspection. Applying still more energy simply creates a bigger black hole.
So in reality, in practice... There is not an infinite number of points on a line between two outstretched fingertips, there are a vast quantity of Planck distances.
Similarly, there is not an infinity of time between 15:03:01 and 15:03:02, there are 5*10^44 moments. Each of which is the amount of time it takes light to travel 1 planck length.
Or so some boffins say, in order to get to sleep at night.