Question

What is the negation of the statement below:

Given a sequence {a_n}, there exists a positive number M, such that |a_n| <= M for any n.

Answer 1

would it just be
M, such that |a_n| > M?

Answer 2

there doesnt exist a positive number M such that |a_n| > M for any any n

Answer 3

NOT(given a sequence a(n), there exists a positive number M such that |a(n)| <= M for any n).

So, given a sequence a(n), there DOES NOT exist a positive number M such that |a(n)| <= M for any n.

Answer 4

you have to flip the equality sign if you're negating the statement

Answer 5

Nope. We're not multiplying by -1. We're just negating the claim.

Answer 6

the negation of x <= 4 is x > 4
in x <= 4, then numbers 1,2,3, AND 4 all work
to negate that, 1,2,3, AND 4 CAN'T work
so to ensure that you must write x > 4

Answer 7

So, here's my take-away. I see your point, but I think -- and correct me if I'm wrong -- you mean to say in your first response:

There DOES exist a positive number M such that |a_n| > M for any any n.

Answer 8

Err, no.

Answer 9

Sorry, I've confused things quite a bit.

Answer 10

You said "there doesnt exist a positive number M such that |a_n| > M for any any n". This seems to just be rephrasing the original claim: "there exists a positive number M such that |a_n| <= M"

Answer 11

okay yeah:
There DOES exist a positive number M such that |a_n| > M for any any n.
would be correct. I just proved it to myself using some predicate logic and law of equivalences.

Answer 12

okay so yall believe that my original conclusion was correct?

Answer 13

yup

Answer 14

thank okay.. if you have a little time, and know some history... please help out with the next question.

Answer 15

ill try