Calculating function convergence
1. A symbolic expression
Consider the very simple, somewhat informal, expression:
2. The exact expression
Expressed exactly according to the wording of the definition of function convergence, the above expression represents the object:
The real to which the function
I would argue that symbolic form is only exact when
The definition is paired with a proof that such a number is unique, so the use of 'the' instead of 'a' is justified.
3. Unwrapping the definition of function convergence.
If we expand the meaning of "converges to <
The real to which the function
"when" was introduced into the wording, as without it
it was not clear that "is ε-close to near
4. We can check, but not compute
The statement in 3 allows us to check if a certain real
For any
5. Using Proposition 9.3.9
What an ordeal... This is why Prop. 9.3.9 is so important. Repeated here it is:
Let
converges to at in .- For every sequence
which consists entirely of elements of and converges to , the sequence converges to .
Using this proposition, we can calculate the value of the expression
in 1. This is because we are able to express everything in terms of limits
of sequences, for which we have already developed a wealth of propositions
covering equality, such as Theorem 6.1.19. Using these theorems, we don't
need to rederive results like I did above for the constant function
6. Utilize sequence limit laws
Taking our expression from 1, Prop. 9.3.9 allows us to replace it with the expression:
The object to which any sequence
And we can ignore the requirements on sequence