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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

x y

. In other words,

[...]

3. Constant multiplication. The sequence

(c a_{n})_{n = m}^{\infty}

converges to

c x

. 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 \neq 0

and

b_{n} \neq 0

for all

n \geq 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 \neq 0

and

b_{n} \neq 0

for all

n \geq 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{l i m_{n \to \infty} a_{n}}{l i m_{n \to \infty} b_{n}}

7. The sequence

(m a x (a_{n}, b_{n}))_{n = m}^{\infty}

converges to

m a x (x, y)

; in other words,

lim_{n \to \infty} m a x (a_{n}, b_{n}) = m a x (lim_{n \to \infty} a_{n}, lim_{n \to \infty} b_{n})

8. The sequence

(m i n (a_{n}, b_{n}))_{n = m}^{\infty}

converges to

m i n (x, y)

; in other words,

lim_{n \to \infty} m i n (a_{n}, b_{n}) = m i n (lim_{n \to \infty} a_{n}, lim_{n \to \infty} b_{n})

06.09.2019