quantum mechanics - Schroeder's Thermal Physics: Multiplicity of a Single Ideal Gas Molecule

I'm reading Daniel Schroeder's An Introduction to Thermal Physics. Please see the below pictures for the text from which I have questions:

1. On pp. 69, I don't get the underlined text. What does he mean by there being a finite number of independent wave functions if both position and momentum space are limited. For, say, a particle in a box, all the wave functions of the TISE are sinusoidal of the form - $\sin \Bigg ( \displaystyle \frac{2 \pi n x}{\displaystyle L} \Bigg )$. They're all orthogonal but the set of all such functions is countably infinite, no?



2. More Importantly I am not able to understand his description of the number of distinct position states and how he uses it to calculate the multiplicity of a single monoatomic ideal gas in a container at fixed energy and volume. I don't see the rationale/reasoning behind. This derivation proceeds to the first few lines of pp. 70 as well.

Please ignore the highlighted text toward the end of pp. 70.

Thanks.