Upper Bound

Upper and Lower Bound

In this video, I define what it means for a set to be bounded above and bounded below. This will be useful in our definition of inf and sup.

[Nikhil_Kumar_Math]

Love the way Proff Peyam provides the motivation of the next topic at the end of each lecture which helps form links between the various topics!

[pushprajbhardwaj]

Thanksgiving sir.

Much love from India 😉❤❤❤

[davidgould]

Nearly made an arse of myself by commenting before watching the end that, by the definition, anything 4 and over would be an upper bound, and wondering what was going on there (then thought: that applies to all upper bounds, whether within the set or not). Then the question was nicely "raised" (sorry, DrP - it's me again) and all was explained. Love the vids.

[karanm]

Hello Sir!! Awesome videos as always!!

[iabervon]

It strikes me that the reason it's worth defining upper and lower bounds, as opposed to jumping straight to suprema and infima, is that it's sometimes easier and sufficient to find bounds that aren't even close to the set. For example, you could define limits as: The limit of f as x goes to c is L if, for all epsilon > 0, there is a delta > 0 such that {f(x): x in (c-delta, c+delta)-{c}} is bounded by L-epsilon and L+epsilon. In a particular proof, you generally couldn't make both bounds tight, and you don't bother to make either bound tight.

[shayanmoosavi]

That was a cool looking "m" right there :)
Almost looking like 👌from front (with missing thumb and index) or 🤘(with missing pinky) :)

Rock on🤘🤘and keep up the good work👌

[Apollorion]

I have a question: "If A and B are both an empty subset of the complex numbers, can there be, beside their labels, any difference between A and B?"

[andywright]

Given that we are talking about real numbers, with arbitrarily long decimal expansions, since a vanishingly small fractipn have finite number of decimal places, does M actually exist?