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Math and science::Analysis::Tao::09. Continuous functions on R

# Function convergence's equivalence to sequence convergence

Let $$X$$ be a subset of $$\mathbb{R}$$, let $$f : X \to \mathbb{R}$$ be a function, let $$E$$ be a subset of $$X$$, let $$x_0$$ be an adherent point of $$E$$, let $$L$$ be a real, and let $$\varepsilon > 0$$ be a real. Then the following two statements are logically equivalent:

• $$f$$ converges to $$L$$ at $$x_0$$ in $$E$$.
• For every sequence $$(a_n)_{n=0}^{\infty}$$ which consists entirely of elements of $$E$$ and converges to $$x_0$$, the sequence $$(f(a_n))_{n=0}^{\infty}$$ converges to $$L$$.

In view of the above proposition, we will sometimes write "$$f(x) \to L$$ as $$x \to x_0$$ in $$E$$" or "$$f$$ has a limit $$L$$ at $$x_0$$ in E" instead of "$$f$$ converges to $$L$$ at $$x_0$$ in $$E$$" or $$\lim_{x\rightarrow x_0;x\in E}^{\infty} f(a_n) = L$$.

### 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.

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