Math and science::Analysis::Tao::06. Limits of sequences

# Limit laws

Let $$(a_n)_{n=1}^{\infty}$$ and $$(b_n)_{n=1}^{\infty}$$ be sequences of reals that converge to the reals $$x$$ and $$y$$ respectively. i.e.
$\lim_{n\to\infty}a_n = x \\ \lim_{n\to\infty}b_n = y$
The 8 basic limit laws are:

1. Addition. The sequence $$(a_n + b_n)_{n=m}^{\infty}$$ converges to $$x + y$$. In other words,
[...]

2. Multiplication. The sequence $$(a_n b_n)_{n=m}^{\infty}$$ converges to $$xy$$. In other words,
[...]

3. Constant multiplication. The sequence $$(c a_n)_{n=m}^{\infty}$$ converges to $$cx$$. In other words,
[...]

4. Subtraction. The sequence $$(a_n - b_n)_{n=m}^{\infty}$$ converges to $$x - y$$. In other words,
[...]

5. Reciprocation. Suppose $$y \ne 0$$ and $$b_n \ne 0$$ for all $$n \ge m$$. Then the sequence $$(b_n^{-1})_{n=m}^{\infty}$$ converges to $$y^{-1}$$. In other words,
$\lim_{n\to\infty}(b_n^{-1} ) = (\lim_{n\to\infty}b_n)^{-1}$

6. Division. Suppose $$y \ne 0$$ and $$b_n \ne 0$$ for all $$n \ge m$$. Then the sequence $$(\frac{a_n}{b_n})_{n=m}^{\infty}$$ converges to $$\frac{x}{y}$$. In other words,

$\lim_{n\to\infty} \frac{a_n}{b_n} = \frac{lim_{n\to\infty} a_n}{lim_{n\to\infty}b_n}$

7. The sequence $$(max(a_n, b_n))_{n=m}^{\infty}$$ converges to $$max(x,y)$$; in other words,
$\lim_{n\to\infty} max(a_n, b_n) = max(\lim_{n\to\infty} a_n, \lim_{n\to\infty} b_n)$

8. The sequence $$(min(a_n, b_n))_{n=m}^{\infty}$$ converges to $$min(x,y)$$; in other words,
$\lim_{n\to\infty} min(a_n, b_n) = min(\lim_{n\to\infty} a_n, \lim_{n\to\infty} b_n)$