I wish to prove that if $(x_n)$ and $(y_n)$ diverge to positive infinity, then $x_n + y_n$ diverges to positive infinity as well.
I started off with definition of divergence. That is: For any real number $M_1$, there exists a real number $N_1$, $n>N_1$ such that $x_n>M_1$. For any real number $M_2$, there exists a real number $N_2, n>N_2$ such that $y_n>M_2$ Now, consider a sequence $(z_n)$ such that $z_n = x_n + y_n$. To prove that it diverges to positive infinity I need to find such $N$, $n>N$ that for any real number $M$, $z_n>M$.
Since $x_n$ and $y_n$ diverge, then $x_n>M_1$ as well as $x_n>M_2$. Similarly, $y_n>M_2$ as well as $y_n>M_1$.
After that, I am stuck as I don't see the way how I can find $N$ for $z_n$.