Proving a limit on a general function

Question

The question is :

Not sure how to even start, i know that |F(x) - L| < Epsilon so if F(x) = L then the function is liner on L, Don't know how to prove its an integer. Thanks in advance !

Answer 1

Note $N$ the integer which has the smallest distance to $L$. As $\lim \limits_{x \to \infty} f(x) =L$, for $\varepsilon =\frac{\vert L-N \vert}{2}$, it exists $b \ge a$ such that for $x \ge b$ you have $$\vert f(x) - L \vert \le \varepsilon =\frac{\vert L-N \vert}{2} \tag{1}.$$ This is true in particular for $x=b$ and $f(b)$ is an integer according to the hypothesis on $f$. This is only possible if $L=N=f(b)$ (if not, $N$ won't be the integer at the smallest distance to $L$). This proves that $L$ is an integer.

As $L=N$, (1) becomes $$\vert f(x) - L \vert \le \frac {\vert L-N \vert}{2}=0$$ for all $x \ge b$. Which proves the second result taking $M=b$.