$\textbf{Convergence of a sequence}$
Let $(a_n)_{n\in\N}$ be a sequence of real numbers and $a\in\R$. The sequence $(a_n)_{n\in\N}$ is said to converge to $a$ if $$\forall\varepsilon>0\space\exists N\in\N\space\forall n\geq N:\space\lvert a_n-a\rvert<\varepsilon.$$
Note: If $(a_n)_{n\in \N}$ converges to $a$, the number $a$ is unique and called the limit of $(a_n)_{n\in\N}$. One commonly writes $\lim_{n\to\infty}a_n=a$ or $a_n\to a$.
$\textbf{Convergence of a sequence}$
Let $(a_n)_{n\in\N}$ be a sequence of real numbers and $a\in\R$. The sequence $(a_n)_{n\in\N}$ is said to converge to $a$ if $$\forall\varepsilon>0\space\exists N\in\N\space\forall n\geq N:\space\lvert a_n-a\rvert<\varepsilon.$$
Note: If $(a_n)_{n\in \N}$ converges to $a$, the number $a$ is unique and called the limit of $(a_n)_{n\in\N}$. One commonly writes $\lim_{n\to\infty}a_n=a$ or $a_n\to a$.
copied
𝐂𝐨𝐧𝐯𝐞𝐫𝐠𝐞𝐧𝐜𝐞 𝐨𝐟 𝐚 𝐬𝐞𝐪𝐮𝐞𝐧𝐜𝐞
Let (𝑎ₙ)ₙ∈ℕ be a sequence of real numbers and 𝑎∈ℝ. The sequence (𝑎ₙ)ₙ∈ℕ is said to converge to 𝑎 if
∀𝜀>0 ∃𝑁∈ℕ ∀𝑛≥𝑁: ∣𝑎ₙ−𝑎∣<𝜀.
Note: If (𝑎ₙ)ₙ∈ℕ converges to 𝑎, the number 𝑎 is unique and called the limit of (𝑎ₙ)ₙ∈ℕ. One commonly writes limₙ→ ͚ 𝑎ₙ=𝑎 or 𝑎ₙ→𝑎.