Tag Archives: sequences

Topology: Sequentially Compact Spaces and Compact Spaces

We’ve arrived at possibly the most confusing notion in topology/analysis. First, we wish to fulfil an earlier promise: to prove that if C is a closed and bounded subset of R and f : R → R is continuous, then f(C) is closed and bounded. [ As … Continue reading

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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

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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

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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

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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

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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

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