Suppose that an ≤b, for all n, and that a=limn →∞ an exists. Show that a ≤b. Conclude that a ≤su

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

Suppose that an ≤b, for all n, and that a=limn →∞ an exists....

Key Concepts

Inequality Preservation Under Limits

A key property in analysis is that if every term of a convergent sequence satisfies a certain inequality with respect to a fixed number, then the limit of the sequence will also satisfy the same inequality. This principle is used to conclude that if all elements of a sequence are bounded by a certain upper bound, the limit of the sequence must also be less than or equal to that upper bound. This concept bridges the behavior of individual sequence elements and their convergent limit.

Supremum

The supremum, or least upper bound, of a set is the smallest number that serves as an upper bound for that set. This concept is central in real analysis because every nonempty set of real numbers that is bounded above has a supremum, according to the Completeness Axiom. It provides a precise way to talk about the 'largest' value that is not necessarily an element of the set itself.

Limit of a Sequence

The limit of a sequence is a fundamental concept in analysis that describes the behavior of the sequence as its index tends to infinity. If a sequence converges, it approaches a unique number called its limit. The definition of convergence ensures that for every small tolerance, beyond a certain index, all sequence elements lie within that tolerance of the limit. This concept is crucial when relating the behavior of an entire sequence to a particular value.

Upper Bound

An upper bound for a set or a sequence in the context of real numbers is a number that is greater than or equal to every element of that set or sequence. Identifying upper bounds is essential to understanding the limiting behavior because if a sequence is bounded above, it ensures that the sequence does not diverge to infinity and allows the application of various convergence theorems.

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