Understanding assumptions wow gold about machine word size in analyzing computer algorithms

I am reading the book Introduction to Algorithms wow items by Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, Clifford Stein. In the second chapter under "Analyzing Algorithms" it is mentioned that :

We also assume a limit on the size of each word of data. For example , when working with inputs of size n , we typically assume that integers are represented by c lg n bits for some constant c>=1 . We require c>=1 so that each word can hold the value of n , enabling us to index the individual input elements , and we restrict c to be a constant so that the word size doesn't grow arbitrarily .( If the word size could grow arbitrarily , we could store huge amounts of data in one word and operate on it all in constant time - clearly an unrealistic scenario.)

My questions are why this assumption that each integer should be represented by c lg n bits and also how c>=1 being the case allows us to index the individual input elements ?

You need at least lg n bits to represent integers of size n, so that's a lower bound on the number of bits needed to store inputs of size n. Setting the constant c >= 1 makes it a lower bound. If the constant multiplier were less than 1, you wouldn't have enough bits to store n.

This is a simplifying step in the RAM model. It allows you to treat each individual input value as though it were accessible in a single slot (or "word") of memory, instead of worrying about complications that might arise otherwise. (Loading, storing, and copying values of different word sizes would take differing amounts of time if we used a model that allowed varying word lengths.) This is what's meant by "enabling us to index the individual input elements." Each input element of the problem is assumed to be accessible at a single address, or index (meaning it fits in one word of memory), simplifying the model.
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