Category Archives: real analysis

Zeno’s Paradoxes

The ancient Greek philosopher Zeno of Elea is known for proposing several paradoxes related to time, space, motion, and infinity.  In this post we’ll focus on one of Zeno’s paradoxes and discuss how some ideas associated with calculus might or … Continue reading

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A Sequence of Right-Hand Riemann Sums Whose Limit Does Not Exist

Recently in my advanced calculus class one of my students asked a (perhaps the) key question when we hit the integration material: Why can’t you just use the limit of the right-hand Riemann sums formed from equally-spaced partitions to define … Continue reading

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Proof of the Recursive Identity for the Bernoulli Numbers

The Bernoulli numbers satisfy the recursive relationship .  (Here, is the Iverson bracket, where evaluates to 1 if P is true and 0 if P is false.)  The relationship can be used to calculate the Bernoulli numbers fairly easily.  This post gives a … Continue reading

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Does Pointwise Convergence to a Continuous Function Imply Uniform Convergence?

Recently in my advanced calculus class we discussed how the uniform limit of a sequence of continuous functions is itself continuous.  One of my students turned the question around: If the limit of a pointwise sequence of continuous functions is … Continue reading

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Proof of the Irrationality of e

In a previous post I proved that is irrational.  In this post I prove the irrationality of e. A proof of the irrationality of e must start by defining e.  There are some different ways to do that.  We’ll take e to be the unique … Continue reading

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Proving the Existence of Irrational Numbers

The ancient Greeks first proved the existence of irrational numbers by proving that is irrational.  The proof is, as modern proofs of irrationality go, fairly simple.  It is often the first example of a proof of irrationality that students see … Continue reading

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Euler Sums, Part II: A Symmetry Formula

A post from a few months ago gave a proof that  In today’s post I’d like to prove a general symmetry formula for Euler sums like this one.  Define and (Notice the upper index on the second sum is , … Continue reading

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