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[hops]

nvm

[hops]

http://imgur.com/S7NUiY2

How does one even "Determine the limit of the sequence using sandwich theorem"? Isn't the entire point of the sandwich theorem to prove limits you already know? YHow would you even use it if you didn't know the limit already?

[Dozebôm Lolumzalìs]

How does one even "Determine the limit of the sequence using sandwich theorem"? Isn't the entire point of the sandwich theorem to prove limits you already know? How would you even use it if you didn't know the limit already?

If you can find two functions such that the sequence is "squeezed" between them, and the two functions have the same limit, the sequence will have that limit.

[MagmaMcFry]

http://imgur.com/S7NUiY2

How does one even "Determine the limit of the sequence using sandwich theorem"? Isn't the entire point of the sandwich theorem to prove limits you already know? YHow would you even use it if you didn't know the limit already?

If it's a weird-looking sequence, but you know intuitively that it behaves like some other, nicer sequences in the limit, then you can use the sandwich theorem to formally prove your intuition right by putting your wacky sequence between two slices of conveniently chosen well-behaved sequence.

[hops]

oh, okay, so all I need to do is eyeball the limit and then find two sequences less or equal and more or equal than the sequence that converges to what i want

[TheDarkStar]

oh, okay, so all I need to do is eyeball the limit and then find two sequences less or equal and more or equal than the sequence that converges to what i want

Find two sequences that converge to the same limit such that for large enough n, one sequence is always smaller than the series whose convergence you want to know and the other sequence is always larger. As n goes to infinity, if the two bounding sequences converge to a value and if your sequence is somewhere in the middle of those two, then your sequence also converges to that value.

[Arx]

The German name, Zwei Polizi und ein Betronken, is much more expressive. If two policemen walk into a cell with a drunk between them, the drunk must end up in the cell.

[Dozebôm Lolumzalìs]

oh, okay, so all I need to do is eyeball the limit and then find two sequences less or equal and more or equal than the sequence that converges to what i want

Sort of. "Eyeballing the limit" doesn't exactly work, but finding the two sequences such that one is always less or equal and one is always more or equal compared to the relevant sequence, and such that the two sequences have the same limit, is sufficient.

The Squeeze Theorem basically says "This sequence is really fucking weird. There's no way I'm going to evaluate that limit. But that's fine! It's stuck between these two nice sequences, these two easy-to-evaluate sequences, that have the same limit. Therefore, the weird-sequence has that limit too."

[Helgoland]

The German name, Zwei Polizi und ein Betronken, is much more expressive. If two policemen walk into a cell with a drunk between them, the drunk must end up in the cell.

That's a good one! I only ever heard 'Quetschlemma' as the German term though.

[MagmaMcFry]

There's also the ham sandwich theorem, which is something else entirely but sounds even tastier. As opposed to, of course, the hairy ball theorem.

[hops]

I've debated on asking this because it looks so brain-dead easy, but I just have no idea how to do any of these. Well, I don't know yet about b) and c) but I just know I can't do a) and the other ones seem dependent on the same premise. I don't really understand how I can really choose a value for the Lipschitz constant. I can prove the Lipschitz continuity. But I just don't know how to choose the constant. I just only know how to prove the constant exists. And yes, I know I can just say it's a real number that approaches infinity but I doubt my teacher will be happy with that answer.

[MagmaMcFry]

For differentiable functions on intervals, any upper bound of |f '| is a valid Lipschitz constant for f and vice versa. So just grab yourself a number that is everywhere larger than |f '| and you're good to go.

[hops]

For differentiable functions on intervals, any upper bound of |f '| is a valid Lipschitz constant for f and vice versa. So just grab yourself a number that is everywhere larger than |f '| and you're good to go.

Why is that?

Also, apparently I was wrong. I have no idea how to even prove Lipschitz Continuity. I know the definition, but damn if I know how to apply it. Can someone explain it to me like I'm an idiot? Because I am an idiot who can't grasp simple mathematic concepts apparently.

[hops]

I'm frustrated and tired and I don't understand why I can't understand anyone's explanation.

[hops]

Look, the textbook examples just show how to provei t for x^2, it really does NOT show me how to do it for literally anything else...