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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,
\[ \lim_{n\to\infty}(a_n + b_n) = \lim_{n\to\infty}a_n + \lim_{n\to\infty}b_n \]

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

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

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

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,
\[ \lim_{n\to\infty} min(a_n, b_n) = min(\lim_{n\to\infty} a_n, \lim_{n\to\infty} b_n) \]

Source

Tao, Analysis I
Chapter 6, p131