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 Aug14 comment Proving that $\mu$ is $\sup S$ (+1) Neat answer! Aug14 comment Proving that $\mu$ is $\sup S$ @PeterTamaroff Done! Aug14 revised Proving that $\mu$ is $\sup S$ added 23 characters in body Aug14 comment Proving that $\mu$ is $\sup S$ My bad! I wrote the definition wrong. I did not realize. Aug14 comment Proving that $\mu$ is $\sup S$ but $1.5$ is not an upper bound Aug14 comment Proving that $\mu$ is $\sup S$ I am sorry, I don't follow Aug14 comment Why is $b^x \overset{\mathrm{def}}{=} \sup \left\{ b^t \mid t \in \Bbb Q,\ t \le x \right\}$ for $b > 1$ a sensible definition? oh, okay thanks Aug14 answered Proving that $\mu$ is $\sup S$ Aug14 comment Why is $b^x \overset{\mathrm{def}}{=} \sup \left\{ b^t \mid t \in \Bbb Q,\ t \le x \right\}$ for $b > 1$ a sensible definition? I'm sorry, but what does $B(x)$ mean in general? Aug14 comment Proving that $\mu$ is $\sup S$ Spot on! :) The question says that $\mu$ is the supremum iff there is an element of $S$ in the interval. But what we just did was started out by assuming that there is no element of $S$ in the interval and proved that if this is the case, then $\mu \ne \sup S$. So, for $\mu = \sup S$, there has to be an element of $S$ in the interval Aug14 comment Proving that $\mu$ is $\sup S$ I could elaborate more if you'd like to Aug14 comment Are all infinities equal? I do not mean to do any self-marketing but if you want to learn about this from the beginning, try out this blog post that was recently wrote by me wp.me/p2aEXv-2N . I am writing a follow up article to this and it will be out very soon Aug14 comment Proving that $\mu$ is $\sup S$ $\lambda \not \in S$ means that $\lambda$ is a upper bound for $S$ which is less than $\mu$. If you remember, one of the properties of the least upper bound is that if there exists a quantity that is less than the least upper bound (say $l$), that quantity has to be in the set under consideration. Or else, $l$ can't be the least upper bound Aug14 comment Proving that $\mu$ is $\sup S$ Assume that for some $\epsilon >0$, $\not{\exists} x \in [\mu -\epsilon , \mu]$, such that $x\in S$. This implies that $\exists \lambda < \mu$ where $\lambda \not \in S \implies \mu \ne \sup S$. So if $\mu = \sup S$ then has to be an element of S in the interval $[μ−ϵ,μ]$ Aug14 revised Proving that $2^{2^n} + 5$ is always composite by working modulo $3$ added 326 characters in body Aug14 revised Find all integer solutions to $7595x + 1023y=124$ edited tags Aug14 comment Find all integer solutions to $7595x + 1023y=124$ Are $x,y \in \mathbb Z$? Aug14 comment How to simplify the following basic equation Do you mean, $\text{originalState} + \Big(\frac{\text{animatedState} \big(100 - 100( \text{finishPos} - x )\big) }{ 100(\text{finishPos} - \text{startPos})}\Big)$ Aug14 answered Proving that $2^{2^n} + 5$ is always composite by working modulo $3$ Aug14 comment Help in understanding integration by changing the variable The "ILATE" mnemonic, as @J.M. mentioned is particularly helpful when you are first starting out but then later on, with practice, you will be able to figure out the way without any mnemonics as such. In fact, I think using the mnemonic, while helpful at the beginning, will hinder you intuitive development when it comes to integrating by parts!