Back again! help is much appreciate as I seem to have found myself stuck and pretty much turned in a blank worksheet to my professor. He says these types of problems will be on our final, and I have no clue where to start.
Define the sequence (x$_n$) in $R$ by x$_0$:= 1 and x$_n$$_+$$_1$:= x$_n$ + 1/x$_n$. Use any relevant theorems about limits of certain sequences and continuity, thusly proving the seq. (x$_n$) is unbounded.
It's pretty obvious that your sequence is strictly increasing so it either is unbounded or convergent. Let assume that $x_n \to l \in \mathbb{R}$ then we have $l=l+\frac{1}{l}$ which is impossible so $(x_n)$ is unbounded
An unusual approach, but it focuses on the logarithmic growth of $x_n$, henec linear growth of $\exp(x_n)$:
Using $\exp t\ge 1+t$ we can show $1\le x_n\le n+1\le \exp(x_n)$ by induction:
The case $n=1$ is clear: $1\le 1\le 2\le e$.
Assume $1\le x_{n-1}\le n\le \exp(x_{n-1})$. Then $$\begin{align}\exp(x_{n})&=\exp(x_{n-1}+\tfrac1{x_{n-1}})\\ &=\exp(x_{n-1})\exp(\tfrac 1{x_{n-1}})\\ &\ge n\cdot (1+\tfrac 1{x_n})\\ &\ge n\cdot (1+\tfrac 1n) \\&=n+1\end{align}$$ and $$x_n=x_{n-1}+\frac1{x_{n-1}}\le n+\frac 11=n+1 $$ and $$ x_n=x_{n-1}+\frac1{x_{n-1}}\ge 1+\frac1n\ge 1$$ We conclude that $\exp(x_n)\to\infty$ nd hence also $x_n\to\infty$
