Show that a sequence having a finite limit point cannot approach \infty

Question

I tried to think how to approach this problem.

Only thing that I can do is to put the finite limit points with \z_n_1\, \z_n_2\ , ... \z_n_k\

And then what should I do for the next step?

Answer 1

Let $(x_n)$ be a real sequence with limit point $x_0 \in \mathbb R$.

Then we have $x_n \in (x_0-1,x_0+1)$ for infinitely many $n$, hence

$(1) \quad x_n <x_0+1$ for infinitely many $n$.

Now suppose that $x_n \to \infty$. Then there is $N \in \mathbb N$ with

$(2) \quad x_n >x_0+1$ for all $n>N$.

But this contradicts $(1)$.