Upper and Lower Sums We illustrated lower sums in class to show that, given a function that is bounded on , where is a refinement of a given partition of . This post shows the analogous graph for upper sums that we did not draw in class. These are our observations about the areas in […]
In this class, the topic of Riemann Integration was introduced by Dr. K. This post will cover the definitions and lemmas that were covered during class. Note that the definitions build on another because we slowly built up to the idea of being Riemann Integrable. 1. A function f is bounded on [a,b] if is […]
By: Abraham Schroeder and Tongzhou Wang Early Calculus In the past few weeks of class we have been rigorously defining and proving all of the rules of calculus. The proofs that we have been learning in class are rooted in the definitions for functional limits and continuity. Strangely enough these proofs and definitions for calculus were […]
In this definitions blog, I will explore the supremum norm in greater detail. First, recall the definition of the supremum norm: If , then we set the supremum norm . for the “uniform size” of and is the “uniform distance” between and . Notice that the word “uniform” comes up in this definition. Recall previous […]
Any point that warrants use of more than one analogy is probably a fairly “muddy” point. In class, we discussed the negation of uniform convergence using a cattle analogy. For uniform convergence the idea is, if each cow is a value of x, and marks the positions of the cows at time as they are […]
Consider the sequence of functions defined by a) Prove that converges uniformly on and find . In order to better examine this, look at , then . The limit of as is 0. This holds similarly for For and , so converges pointwise to Let be arbitrary and choose (which exists by Archimedean Property). Let both be […]
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