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Let $f$ be a contiuous function defined on an open interval $(a, b)$.

My textbook says that if we suppose $\alpha:=\displaystyle\liminf_{x\to b-0}|f(x)|<\infty$, then there is a monotonically increasing sequence $\{x_n\}$ s.t. $\displaystyle\lim_{n\to \infty} x_n=b$ and $\displaystyle\lim_{n\to \infty} |f(x_n)|=\alpha.$

I don't know why. First, I know the definition of $\liminf$ for sequences but I don't know that for functions. (I found the concept of $\displaystyle\liminf_{x\to \infty}f(x)$ but I couldn't find that of $\displaystyle\liminf_{x\to b}f(x)$.)

And why can we make $\{x_n\}$ be increasing ?

I'd like you to explain these things.

RobPratt
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  • See here for the definition of $\limsup$ of functions ($\liminf$ only needs a slight modification) – peek-a-boo Aug 31 '21 at 01:08
  • You can make ${ x_n }$ be increasing because it's a subsequence of an infinite sequence that's bounded above by $b$ (since each $x_n \in (a, b)$). – Robert Shore Aug 31 '21 at 01:29

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