Basic analysis

sqt thread.

idk if a_n is specified to be real or not, but if it isn't explicit then a_n could approach complex infinity

>>9092231
[math] \frac{1}{a_n} \to 0 \implies |a_n| \to \infty [/math]

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>>9092235
Yeah, it's a real analysis text, so it is presumed we are in the reals

>>9092234
Didn't see one on the first few pages, mb

>>9092237
Are you saying that it really is just an issue about positive or negative infinity? That would make sense.

>>9092231
If [math]\lim_{n \,\to\, \infty} \frac1{a_n} \,=\, 0[/math], then for all [math]\varepsilon \,>\, 0[/math], there exists [math]n_0 \,\in\, \mathbf N[/math] such that for all [math]n \,\geqslant\, n_0[/math], [math]\left| \frac1{a_n} \right| \,<\, \varepsilon[/math] i.e. [math]\left| a_n \right| \,>\, \frac1\varepsilon[/math]. That means [math]\left( a_n \right)_{n \,\in\, \mathbf N}[/math] is divergent towards infinity. There is no contradiction here.

>>9092247
Yeah, I see you took a constructive approach. Does that mean my textbook is wrong? lol.

>>9092241
No, I am saying that you saying that [math] a_n \to L [/math] is wrong. If [math] a_n \to L [/math] then [math] \frac{1}{a_n} \to \frac{1}{L} [/math] as long as L is not equal to 0. And if L was zero then the quotient would approach an infinity or not converge at all, instead of approaching 0.

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>>9092256
I should have been more clear, what I gave was a proof to the contrary; that, in fact, the converse of statement (11) in the text is true! And I essentially began my proof by saying, "suppose otherwise." That's why it leads to a contradiction.

It helps to see others think the same way I do about this. I'm guessing it must have been an issue about {a(n)} possibly diverging towards negative infinity instead of positive infinity.

Apologies, I haven't learned latex yet.

>>9092250
>Does that mean my textbook is wrong? lol.
I said "diverges towards infinity," but not in the same context as your textbook. On the complex plane, there is one infinity (in all directions), but on the real line, there are two (positive and negative). The converse your textbook refers to is true on the complex plane, but false on the real line.

Take [math]\left( \left( -2 \right)^n \right)_{n \,\in\, \mathbf N}[/math]. It has no limit on the real line (but you could extract two subsequences that diverge respectively towards [math]+\infty[/math] and [math]-\infty[/math]), yet [math]\lim_{n \,\to\, \infty} \frac1{\left( -2 \right)^n} \,=\, 0[/math]. That's a counter example.

>>9092274
Ohhh, that is tricky... Thank you

There's no such thing as positive and negative infinity - infinity is just infinity.

Mattuck's Introduction to Analysis?

>>9092231
It mentions why in the image itself...
If 1/(a_n) goes to 0, then a_n can go to plus/minus infinity (not just plus).

>>9092397
Yeah, it's a great read so far. Would recommend, although I'm only through chapter 5. He really values the reader's intuition, which is great for an introductory text.

>>9092231
>being a white brainlet

>>9092274
>>9092281
I didn't mention it early, but that was a really helpful response. This idea that something can diverge towards infinity in an absolute sense, yet still have no limit, infinite or otherwise, is still new to me. Thanks a ton, again

>>9092282
There is no such thing as "infinity" either.