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https://www.youtube.com/watch?v=PEQ1-tjlnmo The reverse triangle inequality $$|x-y|ge ||x|-|y||$$ is a classic textbook exercise using the ordinary triangle inequality, but it is also a useful result in calculus or mathematical analysis. We prove it here using two different methods.

https://www.youtube.com/watch?v=mUhBemtfivQ The triangle inequality $$|x|+|y|ge |x+y|$$ for real numbers is an indispensable tool in mathematical analysis, otherwise known as calculus, because it allows us to approximate quantities by bounds. We prove a general version of it using the ordinary absolute value and for any number of variables.

https://www.youtube.com/watch?v=aLbfdbiAkXg Common knowledge says that banks give us a better deal by using compound interest instead of simple interest. Even though there are forces working in both directions, this is indeed true. This fact is equivalent to Bernoulli’s inequality. We also show that the AM-GM inequality can be used to prove that compounding more times

https://www.youtube.com/watch?v=yFPiHQ9fpkA Bernoulli’s inequality is a classic multivariable inequality $$(1+x)^m ge 1+mx$$ that allows us to compare an exponential expression with a linear one. Check out a proof by induction in this video.

https://www.youtube.com/watch?v=4pdI7xBIseM Are the integers bounded above by a real number? Nope. It turns out that for every real number, there exists an integer that exceeds it. We prove this property, called the Archimedean property, using a semi-secret but crucial property of the real numbers called Dedekind completeness.

https://www.youtube.com/watch?v=t_sZk0G_5w4 Sums and products can be written compactly using $sum$ and $prod$ notation, but it is sometimes possible to write the same sum or product in more than one way. The so-called discrete Fubini’s principle is a powerful method of exchanging the order in which the indices occur in a nested sum or product. It

Fractions of the form $frac{1}{n}$ go to zero as $n$ goes to infinity, but what if we add them all up? A surprising classic result is that this sum can get arbitrarily large. We prove the divergence of the harmonic series here.

https://www.youtube.com/watch?v=Of0LRRMoO78 The change of base identity for logarithms is a highly useful technique for dealing with logarithm problems. We prove this identity $$log_b x = frac{log_c x}{log_c b}$$ and then use it to derive a few other identities as corollaries.

When we have square roots in the denominator of a fraction, it can be desirable to transform the fraction into the same number but with a different presentation where the denominator is rational. We show a method for achieving this goal, with the main technique being the difference of squares factorization $$a^2 – b^2 =