Let $f$ be a bounded function and continous in $[a, \infty )$ $M=\sup(f(x)), m=\inf(f(x))$ when $x\in [a, \infty )$ and suppose that $M,m$ are not in image of $f$.
Prove that limit $\lim_{x\to\infty}(f(x))$ does not exist.
2026-04-12 17:50:15.1776016215
Prove that limit does not exist
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Hint:
There are monotone sequences $(x_n)$ and $(x_n')$ such that $f(x_n) \to M$ and $f(x_n') \to m$ as $n \to \infty$. Can $x_n$ and $x_n'$ be nonincreasing or converge to finite numbers?