Question

Show that $\inf _n a_n \leq \liminf _{n \rightarrow \infty} a_n \leq \limsup _{n \rightarrow \infty} a_n \leq \sup _n a_n$.

   Show that $\inf _n a_n \leq \liminf _{n \rightarrow \infty} a_n \leq \limsup _{n \rightarrow \infty} a_n \leq \sup _n a_n$.

Real analysis

N. L. Carothers 1st Edition

Chapter 1, Problem 22

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Step 1

The infimum \(\inf_n a_n\) is the greatest lower bound of the sequence \((a_n)\). The supremum \(\sup_n a_n\) is the least upper bound of the sequence. The limit inferior \(\liminf_{n \to \infty} a_n\) is defined as \(\lim_{n \to \infty} \inf_{k \geq n} a_k\), and Show more…

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Show that $\inf _n a_n \leq \liminf _{n \rightarrow \infty} a_n \leq \limsup _{n \rightarrow \infty} a_n \leq \sup _n a_n$.

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