Properties of increasing sequences Suppose that a sequence {${x_k}$} is increasing. Show that there is some $N$ such that either $x_k \geq 0$ for all $k \geq N$ or $x_k \lt 0$ for all $k \geq N$.  That is, eventually its terms are all non-negative or are all negative.
Any help on the above would be appreciated. Perhaps just a push in the right direction since I'm not quite sure where to start!
 A: Let the basic idea is to show that there are only two outcomes for an increasing function, it either diverges or converges, if it diverges, it does so to $\infty$, if it converges, then the limit will be either positive or negative.
More formally, let $x_n=x_1+\sum \delta_i$. Since $x_i$ is an increasing function, $\delta_i\geq 0$. Therefore, we have two cases:


*

*$\sum_{i=1}^{\infty} \delta_i = \infty$ OR

*$0\leq \sum_{i=1}^{\infty} \delta_i < \infty$


Define $M:=\min\{i: |x_i| \geq 0\}$
Then, we have for Case 1: $x_M+\sum_{i=M+1}^\infty \delta_i= \infty \implies \exists N\geq M:x_M+\sum_{i=M+1}^{N} \delta_i>0\implies x_{N+1}=x_N+\delta_{N+i}\geq x_N>0$. by induction, $x_i>0\;\; \forall i>N$
For Case 2: $-\infty< x_M+\sum_{i=M+1}^\infty \delta_i< \infty $ since the sum of the positive increments is finite. Therefore, $x_M+\sum_{i=M+1}\delta_i<0\implies x_i<0\;\;\forall i$ else we can apply the logic from Case 1 to conclude it will end up in the positive numbers after some point and stay there. 
A: $\textbf{Hint:}$  
If this statement is not true, then for every $N\in\mathbb{N}$ we have that there is a $k\ge N$ such that $x_k<0$ and there is an $l\ge N$ such that $x_l\ge0$.
Now take an $l$ corresponding to  $N=1$, say, and apply this assumption with $N=l$ to show that 
this contradicts the assumption that $(x_n)$ is increasing.
