Homeomorphism between compactification of real line and unit circle.

I'm currently working on a topology assignment which is, unfortunately, due today. As part of that, I need to show that one-point compactification of the real line, $\mathbb{R}\cup\infty$ is homeomorphic to the unit circle. I've come so far to have defined a function $f$ with

$(x,y)=(\cos(2\arctan(t)),\sin(2\arctan(t)))$, which should map the reals onto the unit circle and be bijective and continuous. Now since my domain is compact, if I can show that the unit circle is Hausdorff, I can conclude that my function is a homeomorphism, correct?

However, which topology would I use for that?

Also, I am having a hard time trying to prove that f is surjective. Thought about dividing it up into two functions, from reals to $(-\pi,\pi)$ and then to the unit circle, but that doesn't really seem to work either.

Any thoughts?

Topology is confusing...

(Intuitively, I absolutely see why all this should be the case, however I am struggling with the formal stuff.)

Edit:

Okay I have tried the approach of defining an inverse function, suggested and then proving that it is continuous.

So far I've got:

$$g=\tan\left(\frac{1}{2}\arctan\left(\frac{y}{x}\right)\right)$$

Problem is, that this function is not bijective, so there is something missing (case distinction?) but I can't figure out what...

At least within $\left(\frac{-\pi}{2},\frac{\pi}{2}\right)$ it seems to work, and it's easy to show that combining f and g gives the identity. Any tips maybe?

Cheers

Tom

I don't know why I didn't say the following at the outset. I guess I was just going along with your method.

Rid your proof of trigonometric functions and instead do the following:

$$ t\mapsto (x,y) = \begin{cases} \phantom{\lim\limits_{t\to\infty}} \left( \dfrac{1-t^2}{1+t^2}, \dfrac{2t}{1+t^2} \right) & \text{if }t\ne\infty, \\[10pt] \lim\limits_{t\to\infty} \left( \dfrac{1-t^2}{1+t^2}, \dfrac{2t}{1+t^2} \right) & \text{if }t=\infty. \end{cases} $$

That's a homeomorphism from $\mathbb R\cup\{\infty\}$ to $\{(x,y)\in\mathbb R^2:x^2+y^2=1\}$.

To show that it's surjective, show that $t=\dfrac{y}{x+1}$ (and notice that $\dfrac{y}{x+1}\to\infty$ as $(x,y)\to(-1,0)$ along the curve $x^2+y^2=1$).

It is also useful to know that if X and Y are locally compact Hausdorff spaces and are homeomorphic, then so are their one point compactifications. Another way of looking at the problem would be by considering the steoreographic projection which extends to a map from $\mathbb{R}\cup \{\infty\}$ to $\displaystyle S^1$.