By definition from Wiki, for $f(x)$ a real function, the limit of $f$ as $x$ approaches infinity is $L$ if for all $\epsilon > 0$, there exists a $M$ such that $|f(x) - L| < \epsilon$ whenever $x > M$.
Now I can prove a function $g(x)$ that for all $\epsilon > 0$, there exists a $M$ such that $|g(x) - L| < 2\epsilon$ whenever $x > M$. Can I say $\underset{x \rightarrow \infty}{\lim}~g(x) = L$?