can someone please tell me if my proof of the lemma below is legit. At first I like to give a definition so you know what the lemma is about.
Definition: Let $g:[0,1] \to S^{1}$ be a closed path in $S^{1}\subset \mathbb{C}$ and $\phi: \mathbb{R}\to S^{1}, t \mapsto \exp(2\pi i t)$. We notate the lift of $g$ with $\ f$, which means that $\ f:[0,1]\to\mathbb{R}$ is the unique map that satisfies $\ f(0)=0$ and $g=\phi\circ f$ on $[0,1]$. Then we define the winding number as $w(g):=f(1)\in\mathbb{Z}$
Lemma: Let $g_{0}, g_{1}: [0.1]\to S^{1}$ be two homotopic loops. Then $w(g_{0})=w(g_{1})$ which means that the winding number is well-defined.
Proof: Let $\ f_{0}, f_{1}:[0,1]\to \mathbb{R}$ be the lifts of $g_{0}$ and $g_{1}$ such that $g_{0}=\phi \circ f_{0}$ and $g_{1}=\phi\circ f_{1}$. As $g_{0}$ and $g_{1}$ are homotopic loops we know that the lifts $f_{0}$ and $\ f_{1}$ are also homotopic paths (I'm allowed to use that argument). This gives us a homotopy $H: [0,1]\times [0,1]\to \mathbb{R}$ with $ H(x,t)=f_{t}(x)\\ H(0,t)=f_{t}(0) \text{ and } H(1,t)=f_{t}(1) \text{ for all } t\in [0,1] \\ H(x,1)=f_{1}(x) \text{ and } H(x,0)=f_{0}(x) \text{ for all } x\in [0,1]$.
As $[0,1]\times [0,1]$ is a compact set we know that $H$ is uniformly continious on that set, which means: $\forall \epsilon > 0\ \exists \delta >0\ \forall (x,t)\in [0,1]\times [0,1]:$ $d((x,s),(y,t))<\delta \Rightarrow d(H(x,s),H(y,t))<\epsilon$, where $d$ is the absolute value metric on $\mathbb{C}$ respectively $\mathbb{R}$.
For $x=1$ and $\epsilon=1$ we can find a $\delta>0$ such that: $d((1,s),(1,t))<\delta \Rightarrow d(w(g_{s}),w(g_{t}))=d(f_{s}(1), f_{t}(1))=d(H(1,s),H(1,t))<1$. By the definition of the winding number we know that $f_{s}(1)\in \mathbb{Z}$ for all $s\in [0,1]$, which implies $d(w(g_{s}),w(g_{t}))=0$. In other words: $w(g_{s})=w(g_{t})$ holds for all $s,t \in [0,1]$, i.e. $w(g_{0})=w(g_{1}).$
I kinda dozed off at the end there :). More seriously: all that delta-epsilon stuff really isn't necessary here. There's a more straightforward way to get to the end.
Suppose we define $W(s)$ to be the winding number of the loop defined by $$ \gamma_s(t) = \phi \circ H(s, t). $$ Then evidently $$ W(s) = H(s, 1). $$
Then because $\phi(W(s))$ must be the point $g_0(0)$, we know that $W(s)$ is an integer. Thus $W$ is a continuous integer-valued function. (It's continuous because it's a restriction of a continuous function, $H$, to a subset of its domain). Because the integers are discrete (in the subspace topology), this function must be a constant, so $W(0) = W(1)$, which is, I believe, what you want.
To answer your original question, though: yes, it looks as if this proof is in fact correct, if perhaps overcomplex.