Let $L=\mathbb{Z}^2 \subset \mathbb{R}^2$ be the points with integer coordinates in the plane. Given a point $P \in \mathbb{R}^2$, consider the function $f_P:L \to \mathbb{R}$ defined by $$f_P(X)=d(P,X)=(\text{distance between } X \text{ and }P)^2.$$ Find infinitely many examples of points $P \in \mathbb{R}^2$ such that the distances from $P$ to the elements of $L$ are all different.
I assumed there are infinitely many $P$ as $\mathbb{R}$ itself is uncountable, and since the distance from a point $(x,y)$ is unbounded as $x$ and $y$ increase.
I'm not sure if this is what I'm supposed to do, so yea need help guys. TYVM! BTW, I'm learning proof techniques as of now, I've learnt the injective and bijective thingy. And we were learning matrix and vector before this. THX again!
The locus of points equidistant from two points $X_1$ and $X_2$ is the perpendicular bisector of the line segment $X_1X_2$. Let $\mathcal B$ be the set of all such perpendicular bisectors where $X_1$ and $X_2$ are two different points in $L$. So $\mathcal B$ is a countable set of lines, and we're looking for points which don't lie on any of those lines.
Consider the horizontal line $y=\frac13$, which is not in $\mathcal B$. It's easy to see that any line in $\mathcal B$ is either parallel to that line (so it doesn't intersect it at all) or else intersects it at a rational value of $x$. So the set $$\left\{\left(x,\frac13\right):x\text{ is irrational}\right\}$$ is an uncountable set of points with the property you want.