a) exists a, c ER such that for every n > 0. |an – al < c; b) There is a ER and there is a sequence (an)n>0 of real numbers such that an → 0 and for every n 2 0, lan – a < an; c) (an)n>0 is bounded sequence and for every n > 0, an+2 < : (an + an+1). Which of the conditions a), b) or c) is: -necessary. -sufficient. -necessary and sufficient. for the sequence (an)n>o to converge.
Let $\left(a_{n}\right)_{n \geq 0}$ be a sequence of real numbers and conditions:
a) exists $a, c \in \mathbf{R}$ such that for every $\mathrm{n} \geq 0$. $\left|\mathrm{a}_{\mathrm{n}}-\mathrm{a}\right|<\mathrm{c}$;
b) There is $a \in \mathbf{R}$ and there is a sequence $\left(\alpha_{n}\right)_{n \geq 0}$ of real numbers such that $\alpha_{n} \rightarrow 0$ and for every $n \geq 0,\left|a_{n}-a\right|<\alpha_{n}$;
c) $\left(a_{n}\right)_{n \geq 0}$ is bounded sequence and for every $n \geq 0, a_{n+2} \leq \frac{1}{2} \cdot\left(a_{n}+a_{n+1}\right)$.
Which of the conditions a), b) or c) is:
-necessary.
-sufficient.
-necessary and sufficient.
for the sequence $\left(a_{n}\right)_{n\geq 0}$ to converge.
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