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Tag Archives: convergence
Topology: Nets and Points of Accumulation
Recall that a sequence in a topological space X converges to a in X if the function f : N* → X which takes is continuous at . Unrolling the definition, it means that for any open subset U of X containing a, the set contains (N, ∞] for some N. In … Continue reading
Posted in Notes
Tagged advanced, closed subsets, continuity, convergence, limits, metric spaces, nets, points of accumulation, sequences, topology
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Topology: Limits and Convergence
Following what we did for real analysis, we have the following definition of limits. Definition of Limits. Let X, Y be topological spaces and . If f : X{a} → Y is a function, then we write if the function: is … Continue reading
Posted in Notes
Tagged advanced, continuity, convergence, extended reals, Hausdorff, limits, sequences, topology
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Basic Analysis: Uniform Convergence
Once again, let be a subset. Suppose we now have a sequence of functions , where n = 1, 2, 3, … , such that for each x in D, the sequence converges to some real value. We’ll denote this value by f(x), thus … Continue reading
Posted in Notes
Tagged advanced, analysis, continuity, convergence, pointwise convergence, series, uniform convergence
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Basic Analysis: Sequence Convergence (4)
In this article, we’ll consider the convergence of an infinite sum: . We call this sum an infinite series. Let be the partial sums of the series. Definition. We say that is L (resp. ∞, ∞) if the partial sums converge to … Continue reading
Posted in Notes
Tagged abel transformation, alternating series, analysis, basel series, convergence, series, telescoping series
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Basic Analysis: Sequence Convergence (3)
So far, we’ve been considering the case where a sequence converges to a real number L. It’s also possible for a sequence to approach +∞ or ∞. The infinity symbol “∞” should be thought of as a convenient symbol instead of … Continue reading
Posted in Notes
Tagged analysis, convergence, limit inferior, limit superior, limits, monotone convergence theorem, sequences, squeeze theorem
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Basic Analysis: Sequence Convergence (2)
Monotone Convergence We start with a useful theorem. Monotone Convergence Theorem (MCT). A sequence is monotonically increasing (or just increasing) if for all n. Now the theorem says: an increasing sequence with an upper bound is convergent. Proof. Let L = sup{a1, a2, … }, … Continue reading
Basic Analysis: Sequence Convergence (1)
Much of analysis deals with the study of R, the set of real numbers. It provides a rigourous foundation of concepts which we usually take for granted, e.g. continuity, differentiation, sequence convergence etc. One should have a mental picture of the … Continue reading
Posted in Notes
Tagged analysis, completeness of reals, convergence, infimum, sequences, supremum
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