Problems when proving surjectivity.

I have some question when proving surjectivity of functions in some type of propositions such as this one from Terrence Tao Analysis book (Proposition 3.6.14, (c)):

Let $X$ be a finite set, and let $Y$ be a subset of $X$. Then $Y$ is finite, and $\#(Y ) ≤ \#(X)$.

I am ok with the strategy to prove such statements: We need to construct a bijection from a finite set to Y. Also, since $x$ is finite, there is a bijection $f:\{1,2,...,m\}\to X$ for some natural number $m$. Hence, we should be able to construct a bijection $g$ in terms of $f$.

I have tried two approaches when constructing $g$:

1. defining $g:\{1,2,...,n\}\to Y$ for some $n\leq m$ such that $g(i)=f(i)$ for all $i$ in the domain
2. defining $A=f^{-1}(Y)$ and $g:A\to Y$, such that $g(i)=f(i)$ for all $i\in A$ .

In both approaches, I mannaged to prove injectivity with no problems, but when it came to surjectivity, I was in trouble:

1. surjectivity of case 1:

We want to show that for every $y\in Y$, there is an $i\in \{ 1,2,\dots,n \}$ such that $y=g(i)$.

Let $a\in Y$ be arbitrary. Since $Y\subseteq X$, it follows that $a\in X$. Since $f$ is bijective, there is a unique $j\in \{ 1,2,\dots,m \}$ such that $a=f(j)$.

Now I am in trouble, since $j$ could be greater then $n$. I think I could never have picked a set $\{1,2,...,n\}$ in the first place, because then I am almost saying in advance that the cardinality of $Y$ is $n$, but I'm not sure of this.

2. In my second approach I started the proof like this:

Let $y\in Y$ be arbitrary. Since $Y\subseteq X$, $y\in X$.

Since $f$ is bijective, there is a unique $i\in \{ 1,2,\dots,m \}$ such that $y=f(i)$.

Since $y\in Y$, it follows that $f(i)\in Y$ and so $i\in f^{-1}(y),$ which implies $i\in A$. Thus $g(i)=f(i)=y$.

Thus, $g$ is surjective.

This approach seemed to work, but I don't know if I can assume that $f^{-1}(Y)$ is finite at this point. It seems to already depend on the fact that a subset of a finite set is finite, which is what I am trying to prove...

In summary my questions are:

1. What exactly is wrong with approach 1?
2. Is there a problem with assumption 2? Can I assume that $f^{-1}(Y)$ is a finite set as I did?
3. If none of the approaches work after some adjustments, I need a hint to proceed...