Speaker
DANIEL MARRI
Description
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\textbf{\huge Abstract}
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Elementary analysis dwells so much on the notion of limit of real-valued function or of a sequence in $ \mathbb{R}. $
In spite of the numerous applications of sequences, a sequence may not serve much purpose if its limiting value does not exist. This is because the limit of a sequence is the fundamental notion on which the whole of analysis ultimately rests. For instance, certain basic concepts in Mathematics such as \textbf{continuity, differentiability, integration e.t.c} are defined in terms of limit. Specifically of interest to us is class of sequences that is bounded. The question we pose is this: Is every bounded sequence in $ \mathbb{R} $ convergent? To answer this, we investigate the conditions under which a bounded sequence converges and recommend a criterion for the convergence of a bounded sequence.
For instance the sequence $ a_n = (-1)^n $ is bounded but does not converge. Again, the sequence $ a_n = (-1)^n+ \dfrac{1}{n} $ also fails to converge even though it is bounded. What then accounts for the convergence of a bounded sequence in $ \mathbb{{R}}? $.\\
We therefore conclude that a bounded sequence $a_n$, $n=1,2,3,\cdots $ in $\mathbb{R}$ will converge if any of the following conditions hold:
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\item The sequence $a_n$ is monotonic increasing or monotonic decreasing, that is $a_n$ is monotone.
\item $\lim\limits_{n\to \infty }\inf a_n\;\; =\;\; \lim\limits_{n\to \infty }\sup a_n$ or $ \inf A $ = $ \sup A $, where $A$ is the set of subsequential limits.
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The sequences $ a_n = \dfrac{1}{2^n} $, $n=1,2,3,\cdots $ and $ a_n = \dfrac{1}{n^2} $, $n=1,2,3,\cdots $ are all bounded and convergent since they are all monotone and $\lim\limits_{n\to \infty }\inf a_n\;\; =\;\; \lim\limits_{n\to \infty }\sup a_n$ for all the sequence.
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