Section outline

      • New subsection

        Sequences in Real Analysis A sequence in real analysis is a function from the set of natural numbers into the real numbers, typically denoted as ( π‘Ž 𝑛 ) 𝑛 = 1 ∞ (a n ​ ) n=1 ∞ ​ , where each term π‘Ž 𝑛 a n ​ represents the value of the sequence at the 𝑛 n-th position. Sequences are fundamental to understanding limits, convergence, and the structure of the real number system. Key topics include bounded and monotonic sequences, subsequences, limit points, and the Bolzano-Weierstrass Theorem. Sequences provide the groundwork for more advanced concepts such as series, continuity, and differentiability.

      • Limits of a sequence

        A sequence

        {an}\{a_n\}

        {an​} converges to a real number

        LL

        L if:

        βˆ€Ο΅>0,βˆƒN∈N such that n>Nβ‡’βˆ£anβˆ’L∣<Ο΅\forall \epsilon > 0, \exists N \in \mathbb{N} \text{ such that } n > N \Rightarrow |a_n - L| < \epsilon

        We write

        lim⁑nβ†’βˆžan=L\lim_{n \to \infty} a_n = L

        .
        If no such

        LL

        L exists, the sequence diverges.

        Basic limit theorems:

        • lim⁑(anΒ±bn)=lim⁑anΒ±lim⁑bn\lim (a_n \pm b_n) = \lim a_n \pm \lim b_n

        • lim⁑(anbn)=(lim⁑an)(lim⁑bn)\lim (a_n b_n) = (\lim a_n)(\lim b_n)

        • lim⁑(anbn)=lim⁑anlim⁑bn\lim \left( \frac{a_n}{b_n} \right) = \frac{\lim a_n}{\lim b_n}

          , provided

          lim⁑bnβ‰ 0\lim b_n \neq 0