\(1)\). Every convergent sequence is bounded.
\(2)\). A sequence cannot converge to more than one point.
\(3)\). Every convergent sequence is bounded and has a unique limit.
\(4)\). \(\text{Bolzano- Weierstrass theorem}\) (for sequences)- Every bounded sequence has a limit point.
\(5)\). The set of the limit points of a bounded sequence has the greatest and the least members.
\(6)\). If \(\{b_n\}\) is any sequence, then $$\text{inf} \;b_n \leq \underline{lim}\;b_n \leq \overline{lim}\;b_n\leq \text{sup}\;b_n$$ \(7)\). If \(\{b_n\}\) is any sequence, then \(\underline{lim}\;(-b_n) = - \overline{lim}\;(b_n), \;\text{and}\)
$$-\underline{lim}\;(b_n) = \overline{lim}\;(-b_n),$$
\(8)\). A necessary and sufficient condition for the convergence of a sequence is that it is bounded and has a unique limit point.
\(9)\). A necessary and sufficient condition for a sequence \(\{a_n\}\) to converge to \(l\) is that to each \(\varepsilon > 0\), there corresponds a positive integer \(m\) such that $$|a_n - l| < \epsilon, \quad \forall n \geq m.$$ \(10)\). A bounded sequence \(\{a_n\}\) converges to a real number \(a\) if and only if $$\underline{\lim} a_n = \overline{\lim}{a_n} = a.$$ \(11)\). If \(\{a_n\}\) is a bounded sequence, then
(i) \(\displaystyle \underline{\lim} {a_n} =\) smallest limit point of \(\{a_n\}\), and
(ii) \(\displaystyle \overline{\lim} {a_n} =\) greatest limit point of \(\{a_n\}\).
\(12)\). Every bounded sequence with a unique limit point is convergent.
\(13)\). A neccessery and sufficient condiiton for the convergent of a sequence \(\{a_n\}\) is that, for each \(\epsilon\) there exists a positive \(n\) such that $$|a_{n+k} - a_n | < \epsilon, \; \forall n \geq m, k \geq 1$$ \(14)\). A sequence \(\{a_n\}\) is said to be Cauchy sequence if for each \(\epsilon > 0\), there exists a positive integer \(k\) such that $$|a_{n+k} - a_n | < \epsilon, \; \forall n \geq m, k \geq 1$$ \(15)\). in real numbers, a sequence is convergent iff it is a Cauchy sequence.
\(16)\). If \(\lim a_n = a\) and \(a_n \geq 0,\) then for all \(n\), \(a\geq 0\)
\(17)\). If \(\{a_n\}\) and \(\{b_n\}\) are two sequences such that (i) \(a_n \leq b_n \; \forall n\) and
(ii) \(\lim a_n = a,\;\lim b_n = b, \) then \(a\leq b\)
\(18).\) \(\textbf{Sandwich theroem}\) Let \(a\) be a real number and let \(\{a_n\}\), \(\{b_n\}\), and \(\{c_n\}\) be three sequences such that \[a_n \leq b_n \leq c_n,\] and
$$\lim{a_n} = \lim {c_n}= L.$$ Then, $$\lim {b_n} = L.$$ \(19).\) \(\textbf{Cauchy's First Theorem on Limits}\) Let \( \{a_n\} \) be a sequence of real numbers that converges to a limit \( a \), i.e., \[ \lim_{n \to \infty} a_n = a. \] Then the sequence of arithmetic means \[ A_n = \frac{a_1 + a_2 + \dots + a_n}{n} \] also converges to the same limit \( a \). That is, \[ \lim_{n \to \infty} A_n = a. \] \(20).\) Let \( \{a_n\} \) be a sequence of positive real numbers such that \[ \lim_{n \to \infty} a_n = a > 0. \] Then the geometric mean \[ G_n = \sqrt[n]{a_1 a_2 \dotsm a_n} \] also converges to \( a \). That is, \[ \lim_{n \to \infty} G_n = a. \] \(21)\). \(\textbf{Cesaro's theorem}\) : Let \( \{a_n\} \) and \( \{b_n\} \) be two sequences such that \[ \lim_{n \to \infty} a_n = a \quad \text{and} \quad \lim_{n \to \infty} b_n = b. \] Then, the following holds: \[ \lim_{n \to \infty} \frac{a_1 b_n + a_2 b_{n-1} + \dots + a_n b_1}{n} = ab. \] \(22).\)Let \( \{a_n\} \) be a sequence of positive real numbers such that \[ \lim_{n \to \infty} \frac{a_{n+1}}{a_n}= l, \] exists, then so does \(\lim_{n \to \infty} (a_n)^{1/n} \) and the two limits are equal, i.e \(\lim_{n \to \infty} (a_n)^{1/n}= \lim_{n \to \infty} \frac{a_{n+1}}{a_n} , \) provided the later limit exists.
\(23)\). If \(\{a_n\}\) be a sequence , such that \(\lim \displaystyle \frac{a_{n+1}}{a_n}=l >1\), then $$\lim a_n = \infty$$ \(24).\) A necessery and sufficnet condition for the convergence of a monotonic sequence is that it is bounded.
\(25).\) A monotonic increasing bounded above sequence converges to its least upper bound and monotonic decreasing bounded below sequence converges to its gratest lowe bound.
\(26).\) \(\textbf{Nested Intervals}\) : If a sequence of closes intervals \([a_n,b_n]\) is such that each amily member \([a_{n+1},b_{n+1}]\) is contained in the previous one \([a_n,b_n]\) and \(\lim (b_n - a_n ) = 0\), then there is one and only one point common to all the intervals of the sequence.
\(27)\). \(\textbf{Cantor's Intersection Theorem}\) : Let \( \{C_n\} \) be a sequence of closed, non empty and bounded sets in \( \mathbb{R} \) such that \[ C_1 \supseteq C_2 \supseteq C_3 \supseteq \dots \] then \[ \bigcap_{n=1}^{\infty} C_n \neq \emptyset. \] \(28).\) \(\textbf{Properties of the Algebra of Sequences}\)
Let \( \{a_n\} \) and \( \{b_n\} \) be sequences of real numbers, and let \( c \) be a constant. The following properties hold: If \( \lim a_n = L_1 \) and \( \lim b_n = L_2 \), then \[ \lim (a_n + b_n) = L_1 + L_2. \] \(i).\) If \( \lim a_n = L_1 \) and \( \lim b_n = L_2 \), then \[ \lim (a_n - b_n) = L_1 - L_2. \] \(ii).\) If \( \lim a_n = L \) and \( c \) is a constant, then \[ \lim c \cdot a_n = c \cdot L. \] \(iii).\) If \( \lim a_n = L_1 \) and \( \lim b_n = L_2 \), then \[ \lim (a_n \cdot b_n) = L_1 \cdot L_2. \] \(iv).\) If \( \lim a_n = L_1 \) and \( \lim b_n = L_2 \), provided \( L_2, b_n \neq \) then \[ \lim \frac{a_n}{b_n} = \frac{L_1}{L_2}. \]
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