Math and science::Analysis::Tao::09. Continuous functions on R
Function convergence's equivalence to sequence convergence
Let
converges to at in .- For every sequence
which consists entirely of elements of and converges to , the sequence converges to .
In view of the above proposition, we will sometimes write "
Use limits of sequences to calculate limits of functions
This proposition is very important in allowing us to determine what real a function converges to without having to manipulate the definition of function convergence and play around with ε-closeness. Instead we can use all of our built up knowledge of limits of sequences.