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Author Archives: limsup
Basic Analysis: Limits and Continuity (2)
Previously, we defined continuous limits and proved some basic properties. Here, we’ll try to port over more results from the case of limits of sequences. Monotone Convergence Theorem. If f(x) is increasing on the open interval (c, a) and has … Continue reading
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Tagged advanced, analysis, continuity, limits, monotone convergence theorem, points of accumulation, squeeze theorem
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Basic Analysis: Limits and Continuity (1)
[ This is a continuation of the series on Basic Analysis: Sequence Convergence. ] In this article, we’ll describe rigourously what it means to say things like . First, we define a punctured neighbourhood of a real number a to be … Continue reading
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
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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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Topics in Commutative Rings: Unique Factorisation (3)
Example 1: The Gaussian Integers Z[i] Let’s pick the norm function N : Z[i]-{0} → N where N(a+bi) = (a+bi)(a–bi) = a2+b2. We know that N is a multiplicative function, i.e. N(r)N(s) = N(rs). Instead of checking this by brute force, we write N(x) = x·xc, where (a+bi)c = a-bi is the conjugate of a+bi. It’s easy to … Continue reading
Topics in Commutative Rings: Unique Factorisation (2)
In the previous article, we imposed certain finiteness conditions on the ring (specifically a.c.c. on principal ideals: that every increasing sequence of principal ideals is eventually constant), then proved that unique factorisation holds if and only if all irreducible elements … Continue reading
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Tagged commutative rings, euclidean domains, irreducibles, prime ideals, primes, principal ideal domains, ring theory, rings, UFDs, unique factorisation
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Topics in Commutative Rings: Unique Factorisation (1)
Unique Factorisation: Basics Throughout this post, let R be an integral domain; recall that this means R is a commutative ring such that whenever ab=0, either a=0 or b=0. The simplest example of an integral domain is Z, the ring of integers. What’s of interest to … Continue reading
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Tagged commutative rings, irreducibles, prime ideals, primes, ring theory, rings, UFDs, unique factorisation
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Introduction to Ring Theory (8)
Matrix Rings In this post, we’ll be entering the matrix. Let R be a ring. The ring Mn×n(R) is the set of matrices whose entries are elements of R, where the addition and multiplication operations are given by the usual matrix addition … Continue reading
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Tagged advanced, cramer's rule, determinants, matrix rings, ring theory, rings, simple rings
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Mathematics and Such 