How can I find an approximation of a smooth function $R^2 \to R^4$ such that when restricted to $S^1$ it is an immersion? I am having trouble with the following exercise:
Given a smooth function $f: R^2 \to R^4$ and $S^{1}$ embedded in $R^2$, then $\forall \epsilon >0$ there exists a smooth function $f_{1}$ such that $sup|f(x)-f_{1}(x)|< \epsilon$, which is an immersion when restricted to $S^1$.

We can do this without invoking transversality. There is a standard theorem that specializes to your case: 
Let $f : \mathbb R^n\to  R^p$ be a smooth map with $2n \le p.$ Then for any $\epsilon > 0$ there exists a $p \times n$ matrix $A = (a_{ij} )$
such that $|a_{ij} | < \epsilon$ and the map $g : \mathbb R^n\to \mathbb R^p$  defined by $g (x) = f (x) + Ax$ is an immersion.
To prove this, first note that $M_k (p, n)\subseteq  M (p,n) $, the set of all $p\times n$ matrices of rank $k$, is a submanifold of $M (p,n) $ of dimension $k (p + n − k).$
We need to find a matrix $A$ such that $\text{rank}\ dg(x)=n.$ Let $S=\{A_{p\times n}:|a_{ij}|<\epsilon|\}.$ Then, $m(S)=(2\epsilon)^{pn}>0.$ Consider $S_1 = \{A \in S : A = B − df (x);\  \text{rank}\  B < n\}.$ The result will follow if we can show that $m(S_1)=0.$
Define $F_k : M_k (p, n) \times \mathbb R^n \to  M (p, n)$ by  $F_k (B, x) = B − df(x)$ and $\Phi (k) = k (p + n − k) + n.$ Then $\Phi'(k) = −2k + p + n > 0$, so $\Phi$ is increasing. This implies that $k (p + n − k)< \text{dim}\ (M (p, n))$ and an application of Sard's theorem shows that $m\left(F_k(M_k (p, n) \times \mathbb R^n)\right)=0$, and so $m(S_1)=0$. 
