A real sequence is a function whose domain is the set of positive integers and range is a subset of the set of real numbers, i.e., $f: N \longrightarrow R$

Example:

Let $f(n)=\dfrac{1}{n}, \forall n \in N$ , then $f$ is a sequence in $R$ .

Limit of Sequence:

Definition:

We say that a sequence of real numbers $(a_n)^\infty_{n-1}$ converges to a real number $l$ if for every $\in > 0$ there is an $N_0$ such that $|a_n-l| < \in, \forall n > N_0$ i.e., a sequence $(a_n)^{\infty}_{n-1}$ converges at all it has to converge to a unique real number $l$ which we call its limit, we express this by

$\lim_{n \to \infty} a_n=l$

where the symbol $'\longrightarrow '$ stands for approaches or tends to.

Properties:

If $\lim_{n \to \infty} a_n=l_1$ and $\lim_{n \to \infty} b_n=l_2$ then,

$\lim_{n \to \infty} (a_n+b_n)=l_1+l_2=\lim_{n \to \infty} a_n+\lim_{n \to \infty} b_n$
$\lim_{n \to \infty}(a_n-b_n) =l_1-l_2=\lim_{n \to \infty} a_n-\lim_{n \to \infty} b_n$
$\lim_{n \to \infty} (a_n \cdot b_n)=l_1\cdot l_2=\lim_{n \to \infty} a_n \cdot \lim_{n \to \infty} b_n$
$\lim_{n \to \infty} \left( \dfrac{a_n}{b_n} \right)=\left( \dfrac{l_1}{l_2} \right)=\dfrac{\lim_{n \to \infty}a_n}{\lim_{n \to \infty}b_n }$

Where $\lim_{n \to \infty} b_n \neq 0$

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