Limit Superior, Limit Inferior, and Convergence Properties of Bounded Sequences
1. Let $\{x_n\}$ be a bounded sequence of non-negative numbers such that for every $\epsilon > 0$, we have
$$ \limsup x_n < \epsilon.$$
Prove that $x_n \to 0$.
2. For a bounded sequence $\{x_n\}$, prove that
$$\liminf x_n = x$$
if and only if for every $\epsilon > 0$, infinitely many terms of $\{x_n\}$ are less than $x + \epsilon$, and eventually, all terms are greater than $x - \epsilon$.
3. Let $\{a_n\}$ and $\{b_n\}$ be two sequences of positive numbers such that $\{a_n\}$ is bounded and $\{b_n\}$ converges. Prove that
$$\limsup (a_n b_n) = (\limsup a_n) \cdot (\lim b_n).$$
Tpolo
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Savionf
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The answer is accepted.
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