What does it mean to say that a sequence $x_n, n=1,2,3,\ldots$ of real numbers is bounded? And also, following on from the question in the title. 
Suppose $x_n$, $n = 1,2,3,\ldots$ is a sequence of positive real numbers that does not tend to infinity. Explain why it must be possible to construct a sub-sequence of $x_n$ that is bounded.
Please bear in mind I'm new to analysis, so adjust your answers accordingly. 
Thanks
The full question can be found here - http://imgur.com/Gz4cCKb
 A: The sequence $-1,-2,-3,\ldots$ has no bounded subsequence, but it does not tend to $+\infty$.
The proposition asserted makes sense if instead of $\pm\infty$ one has just one $\infty$, which is approached by sequences like $+1,-2,+3,-4,\ldots$, and that certainly makes sense in some contexts.
If one does that, then "tends to $\infty$" means for every $M>0$, not matter how big, the sequence eventually gets out of the interval $[-M,M]$ and subsequently stays out of that interval.
"Does not tend to infinity" therefore means that there is some number $M>0$ such that no matter how far you go down the sequence, you will at some time at or after that point return to the interval $[-M,M]$.
That gives you a bounded subsequence.
A: If $x_{n}$ doesn't tend to $ + \infty$ , by definition of convergence to $+ \infty$ there exists $ M > 0$ such that for every $k \in \mathbb{N} \ \ \exists n_{k} > k $ with $$x_{n_{k}} \in [0,M]$$ So you have a bounded subsequence.
A: A sequence of numbers $x_n$ is bounded if there is some number $N$ such that $|x_n|<N$ for all $n$ (equivalently, $-N\leq x_n\leq N$). In other words, a sequence is bounded if we can find a number that is bigger than all of its terms and a number that is smaller than all of its terms. For example, the sequence $x_n=\sin(n)$ is bounded because $-1\leq\sin(n)\leq 1$ for any $n$, but the sequence $x_n=n^2$ is not bounded (why?).
To show that a sequence that does not tend to infinity has a bounded subsequence, try proving the contrapositive: if every subsequence is unbounded then the sequence tends to infinity.
