Baby Rudin ch9 q12d
Fix two real numbers $a$ and $b$, $0 <a < b$. Define a mapping $f = (f_1,f_2 , f_3)$ of $\mathbb{R}^2$ into $\mathbb{R}^3$ by
$f_1(s, t) = (b + a \cos s) \cos t$
$f_2(s, t) =(b + a \cos s) \sin t$
$f_3(s, t) = a \sin s$.
Describe the range $K$ of $f$. (It is a certain compact subset of $\mathbb{R}^3$.)
(d) Let $\lambda$ be an irrational real number, and define $g(t) = f(t, \lambda t)$. Prove that $g$ is a 1-1 mapping of $\mathbb{R}^1$ onto a dense subset of $K$.
I proved 1-1.
Let $L=\{(t, \lambda t)\mid t\in \mathbb{R}\}$
To prove $img\, g$ is dense in $K$.
i.e. to prove $f (L)$ is dense in $K$
attempt 1: $B\subset A$ is dense in $A$ and $f$ is continuous on $A$ $\implies$ $f(B)$ is dense in $f(A)$.
But $L$ is not dense in $\mathbb{R}^2$ $\implies$ this attempt fails.
attempt 2: To prove $\forall \, \epsilon >0, \, \forall \, (s, t) \in \mathbb{R}^2, \, \exists \, t' \in \mathbb{R}$ s.t. $d(f(s,t), f(t', \lambda t'))< \epsilon$. This is probably tough.
1-1ness is not used yet.
Please give a hint. Please do not give solution. Thanks!