If an ≤bn for all n, and if (an) converges, show that limn →∞ an ≤lim infn →∞ bn.

Name: Problem 30, Chapter 1: Real analysis
Uploaded: 2021-11-12T02:32:44-08:00
Duration: 1 min 49 s
Channel: Nick Johnson
Description: If an ≤bn for all n, and if (an) converges, show that limn →∞ an ≤lim infn →∞ bn.

step 1 Okay, so here we are given that A sub N approaches L. So in other words, we have the limit as N

If an ≤bn for all n, and if (an) converges, show that limn...

Key Concepts

Sequence Convergence

This concept refers to a sequence approaching a specific value as the index tends to infinity. When a sequence converges, it means that for any given small tolerance, there exists a point from which all subsequent elements of the sequence lie within that tolerance of a particular number—the limit. Convergence is fundamental in analysis as it ensures the predictable behavior of sequences in the limit.

Limit Inferior

The limit inferior, or liminf, of a sequence is the greatest lower bound of the set of accumulation points (or subsequential limits) of the sequence. It is defined as the limit of the infima of the tail ends of the sequence, offering a way to capture the lower limiting behavior of sequences that might not fully converge. This concept is vital in understanding the behavior of sequences that oscillate or do not have a unique limit.

Order Preservation Under Limits

This concept exploits the property that if one sequence is element-wise less than or equal to another, then the order is preserved when taking limits or limit inferiors, provided the limits exist. In this context, if a sequence converges and is bounded above by another sequence that may not converge, the limit of the convergent sequence provides a lower bound for the limit inferior of the other. This property is key in proofs that involve comparisons between convergent and non-convergent sequences.

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