Does this expression represent the largest real number? I'm not very good at this, so hopefully I'm not making a silly mistake here...
Assuming that $\infty$ is larger than any real number, we can then assume that:
$\dfrac{1}{\infty}$ is the smallest possible positive real number.
It then stands to reason that anything less than $\infty$ is a real number. Therefore, if we take the smallest possible quantity from $\infty$, we end up with:
$\infty-\dfrac{1}{\infty}$
Does that expression represent the largest real number? If not, what am I doing wrong?

You could use the same reasoning to say that since $1$ is a real number then $\infty - 1$ has to be a real number. So you see that if you subtract anything from $\infty$ it does NOT become a real number. Actually $\lim_{x \to \infty} x-1 = \infty$.
To visualize this: Given that you have infinite money and spend 1 dollar. How much money have you left, yes its still infinite. This is a bit hard to understand when starting to deal with infinity, you can for example split your money up into two parts and both parts will remain infinite.
You see that infinity cannot be made finite by subtracting some finite number.
