Is Y Is Inverse Of Every Real Number?

We know that the statement

\(\lim\limits_{x\to \infty}f(x)\) exsixts has the following meaning.Negate the statement. \(\exists(\forall A)(\forall t > 0) (\exists \delta > 0 ) (\forall x) ( 0 < |x – a| < \delta \Rightarrow |f(x) – A| < \varepsilon\)

Solution:

The negative of the statement is

\[\forall(\exists A) ( \exists t > 0) (\forall \delta > 0) (\exists x) (0 < |x – a| < \delta \Rightarrow |f(x) – A| < \varepsilon\]

Write the following statement in symbol form and then form its negation, for each \(\varepsilon > 0\), there exsists an N > 0 s.t \(| fn(x) – f(x) | < ϵ \ Whenever\ n > N and\ S x \in S.\)

Solution:

\[\forall \varepsilon > 0,\ \exists N > 0\] \[s.t\ |fn(x) – f(x) | < \varepsilon\]

Negation:

\[\exists \varepsilon > 0, \ \forall N > 0 \\ s.t\ |fn(x) – f(x) | > \varepsilon\\ \ whenever\ n > N \ and\ x \in S.\]