Question: Does the function $f(x)=\{1$ if $x\in \Bbb Z $, $0$ if $x \notin \Bbb Z$, tend to zero as $x$ tends to infinity?
Not sure how to do the piece-wise function, feel free to edit or tell me how to. I've used the comma instead to illustrate a new line.
Defintion for a function $f(x)$ which tends to $0$ as $x$ tends to infinity:
$\forall \varepsilon > 0 \exists K \in \Bbb R \forall x>K :|f(x)|<\varepsilon$
My claim: Does not tend to zero as $x$ tends to infinity. The logic behind my claim is that if I were to imagine a graph of this, there would be two dotted lines, one at $y=0$ and another at $y=1$.
I find proving questions like these quiet difficult. I usually start off by writing what I need to prove, in this case I need to prove the negation. Then I would proceed with finding my choices for $\varepsilon$ and $x$, this is where I am struggling at the moment.