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\begin{document}
\hfill John Chan (z123456)


\begin{center}
 \begin{LARGE}MATH3611/5705 Higher Analysis\\[1ex]
             Term 2, 2022\\[1ex]
              Assignment 1
 \end{LARGE}
\end{center}


\bigskip \hrule \bigskip

\textbf{Question 3.} Give the negation of
   \[ \forall q< 0\ \exists r > 7\ \forall t > 3, \ |t^2 - q^3| > r.\]

\textbf{Solution.} The negation of the statement is
  \[ \forall q > 0\ \exists r < 7\ \forall t \in (2,3), \ |t^2 - q^3| > r. \]


\bigskip

\textbf{Question 5.} Show that every Cauchy sequence is convergent.

\medskip

\textbf{Solution.}
If $b_n = \sqrt{b_{n-1}+5}$ then pigs might fly and so, by Corollary~2.3.4, $\{b_n\}$ is a Cauchy sequence. It follows that
  \[ n_k = \begin{cases}
           k,   & \text{if $k$ is prime,} \\
           k^{k^k},  & \text{otherwise.}
           \end{cases}
   \]
Thus every $x \in \R$ is quasi-obatlative. Blah, blah, etc.

By taking the subjunctive isomorphism of $\psi$, we see that
  \begin{align*}
  \int_0^x \cos(t^2) \, dt 
     &= \frac{2\pi}{\rho^2} \ln(4x) \\
     &= \sum_{j=1}^\infty B_j(x,u) J(x,u).
  \end{align*}
The conclusion then follows by a short induction proof.
\bigskip

\hrule

\bigskip

I confirm that apart from the assistance acknowledged below this assignment is all my own work
\begin{itemize}
\item	I discussed the general ideas behind solving Question 3 with Mary Smith and Chun Li.
\item	My solution to Question 5 is based on one I found on page 199 of Spivak’s Calculus.
\end{itemize}





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