List of basic formulas and concepts related to Cauchy limits:
- Definition of the Cauchy limit:
- A sequence (a_n) converges to a limit (L) if for any (\epsilon > 0) there exists a natural number (N) such that for all (n > N) the inequality holds:
[
|a_n – L| < \epsilon
]
- Cauchy criterion for sequences:
- A sequence (a_n) is convergent (i.e. has a limit) if and only if for any ( \epsilon > 0) there exists a natural number (N) such that for all (m, n > N) the following holds:
[
|a_n – a_m| < \epsilon
]
- Properties of limits:
- If ( \lim_{n \to \infty} a_n = L ) and ( \lim_{n \to \infty} b_n = M ), then:
- Sum: ( \lim_{n \to \infty} (a_n + b_n) = L + M )
- Product: ( \lim_{n \to \infty} (a_n \cdot b_n) = L \cdot M )
- Quotient: ( \lim_{n \to \infty} \frac{a_n}{b_n} = \frac{L}{M} ) (for ( M \neq 0 ))
- Function limits:
- If a function ( f(x) ) has a limit ( L ) as ( x \to a ), then:
[
\lim_{x \to a} f(x) = L
]
- Theorem on the limit of a composite function:
- If ( \lim_{n \to \infty} a_n = L ) and ( f ) is continuous at the point ( L ), then:
[
\lim_{n \to \infty} f(a_n) = f(L)
]