I am taking an introduction to real analysis class. In class, my professor proves the maximum of a non-empty finite subset of $\mathbb{R}$, if exists, is equal to the supremum in the following way: Say we have $S$ is the subset meets all the requirements. Let $s_1=\max S, s_2=\sup S.$ First part: $s_1$ is the element of $S$, then $s1\leq \sup S=s_2,$ thus $s_1\leq s_2.$ Second part: $s_1&gt;s$, for all, $s \in S$, then $s_1$ is an upper bound for $S$, thus $s1\geq s_2$. Therefore, $s_1=s_2.$  I understand everything about this proof except the end of the second part. How can we conclude $s_1\geq s_2$ based on the fact s1 is an upper bound? Isn't $s_1$ in the subset, whereas $s_2$ can be in and out? The definition of upper bound says that this upper bound is greater and equal to all the elements in the subset. So do we just assume $s_2$ in the subset?  Sorry for my formatting, I am pretty new to this platform and still learning. Thank you for your help in advance.
Prove that given $A\subseteq\mathbb{R}$, if has a maximum $a_0$, then $a_0$ must be the supremum.   In order to prove this, I have written the following:  Let $A\subseteq \mathbb{R}$, nonempty set because $a_{0}$ $\in A$ for hypotesis. Then, since $a_{0}$ $\in A$ is the largest element of the set $A$   (i.e. for each element $A$ of A we have that $ a\leq a_{0}$) $A$ is bounded above. Then, A has a supremum. Let $s$ be the least upper bound of $A,$ so we have that for each element $a$ of $A$, $a\leq s$, for being $s$ an upper bound. In particular, $a_0\leq s$, which implies that $s=a_0$.   Then, we have that $\sup(A)=\max(A)=a_0$, which is exactly what we wanted to prove.  Observation: this proof is true if $s \in A$   I'm not sure if my proof is correct. I will be thankful if someone tells me.
Functional equations $f(x+y)= f(x) + f(y)$ and $f(xy)= f(x)f(y)$ Let $f:\mathbb{R}\to \mathbb{R}$ is a function such that for all real $x$ and $y$, $f(x+y)= f(x) + f(y)$ and $f(xy)= f(x)f(y)$, then prove that $f$ must be one of the two following functions:   $f:\mathbb{R}\to \mathbb{R}$ defined by $f(x)=0$ for all real $x$   OR   $f:\mathbb{R}\to\mathbb{R}$ defined by $f(x)=x$ for all real $x$   I got to the point where putting the two equations together, you get $f(x+y)f(x)= f(xy) + f(x)^2$ and plugging in $f(x)=x$ checks with it. So am I going in the right direction or am I just doing some guess work? Is there a more elegant way of doing it?   Thanks Windows 7 msconfig lets users select the options No GUI boot and OS boot information to have a list of drivers being loaded during boot. Is it possible to have similar detailed messages during shutting down/restarting/logging off? 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I have sent emails to customercare@vfsglobal.com, info.vfsnewyork@vfsglobal.com, NYscanhub.escalate@vfsglobal.com, DLUKVInewyorksensitive@fco.gov.uk and I have received no response from anyone.  Frustrated. I do not know what to do. Please help me!! Why is pressure in a liquid the same in all directions? I'd like answers both in the more intuitive side an on the more precise side.   Thinking of water as "cubes" of water, for example, would allow pressure in the z axis to be independent of the y or x axis. Choosing other "shapes", like a  triangular prism, yield different results. Will this fact, then, be dependent on the format of the molecules of the liquid ?  I've heard of the rotational symmetry of liquids. What is its precise statement ? Why is it true ? $f(x,y)=4x^3y^2$ Dealing with Directional Derivatives and Vectors Let $f(x,y)=4x^3y^2$  How do I find the directional derivative of $f$ at $(2,1)$ in the direction of the vector $3i-4j$? 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I have tried to find the exact way to calculate the focal length of the lens I need but I have not been able to find a good resource to make this simple enough for my limited optics knowledge. Can you please explain a simple way to calculate the focal length of a lens given the following inputs as an example:    I would prefer a simplified mathematical explanation so that I may use it to make similar determinations in the future using a "" method.  I have . Using that method, I came up with:  Required lens focal length = 7.02mm sensor height x (77in / 36in) = 15mm   Is that correct? 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Is it just a glorified folder and a holding place for soon to be deleted files, or does it do something specific?  Specifically, are files that are "moved" to the Recycle Bin actually moved on the hard disc, or are just the pointers to the files moved?  I am a fairly experienced user, I just wanted a more depth explanation of the Recycle Bin. How can the petrification from the Flesh to Stone spell be removed? Add manually installed application to 'Show Applications' grid in Ubuntu 18.04 Root filling up although it should have some space This has been perplexing me for some time. I'm on Ubuntu 11.04. My root partition is filling up and I can't figure out what's causing it. Even stranger, when I run df -h, this is the output:  Filesystem            Size  Used Avail Use% Mounted on /dev/sda2             9.2G  8.8G     0 100% /   As you see, there should be about 400 MB free, but Avail shows 0. What could be causing this? 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