Upper bound for ratio as a function of terms on the LHS

Question

I've recently been stumped over the following inequality. Suppose $x,y \ge 0$ and $a > 1.$ Do there exist constants $c_1, c_2, c_3 \in [0,\infty)$ which do not depend on $x$ or $y$ such that $$\frac{a + \frac{x}{2y}}{a-\frac{1}{2}} \le c_1x + c_2y + \frac{c_3}{y}$$

Thanks in advance for your help.

Answer 1

Consider the sequence $(x_n,y_n)=(n,\frac{1}{n})$. Plugging in gives you:
$$\frac{a + \frac{n^2}{2}}{a-\frac{1}{2}} \le c_1n + c_2\frac{1}{n} + c_3n$$ Sending $n\to \infty$ yields a contradiction (why?).