# Tagged Questions

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### Proof that $\limsup (a_nb_n)=\limsup a_n \lim b_n$

Let $a_n$ be a bounded sequence and let $b_n$ be a convergent sequence. Prove that $\limsup⁡〖(a_n b_n )=\lim sup⁡〖a_n 〗 \lim⁡b_n 〗$. Here is what I got Proof Let $a_n$ be a bounded sequence and ...
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### Limit superior inequalities proof

Let $a_n$ be a positive sequence. Prove that $$\limsup_{n\to \infty} \left(\frac{a_1+a_{n+1}}{a_n}\right)^n\geqslant e.$$
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### Proof for limit superior's property: $\limsup (a_n b_n ) \leq \limsup a_n \cdot \limsup b_n$ [duplicate]

Let $a_n,b_n>0$ for all $n\in\mathbb N$. Prove that $\limsup (a_n b_n ) \leq \limsup a_n \cdot \limsup b_n$ I know that $\limsup (a_n+b_n ) \leq \limsup a_n + \limsup b_n$. But I don't know how ...
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### limit superior and limit inferior proof

$$\limsup \left(\frac 1{a_n} \right)=\frac 1{\liminf(a_n )}$$ I know this is true base on the definition of $\limsup$ and $\liminf$, but I don't know how to prove it formally.