# Problem on a set which is a subset of a finite set

issacnewton
Homework Statement:
Prove that if ##n \in \mathbb{N}## and ##A \subseteq I_n##, then ##A## is finite and ##|A|\leq n ##. Furthermore, if ##A \ne I_n##, then ##|A| < n##. We are given that
$$I_n = \big \{ i \in \mathbb{Z}^+ | i \leq n \big \}$$
Relevant Equations:
Definition of a finite set and definition of bijections
Here is my attempt. Since we have to prove that ##A## is finite, we need to prove that there exists some ##m \in \mathbb{N}##, such that there is a bijection from ##A## to ##I_m##. And hence we have ##A \thicksim I_m##. Now, since there are ##n## elements in ##I_n##, number of elements in ##A## are less than or equal to ##n##. So, we have ## |A|\leq n ##. Let, ##|A| = m##. Now, we can construct a function from ##A## to ##I_n##, such that ##m## elements in ##A## are paired with first ##m## members in ##I_n##. And, we see that the first ##m## members in ##I_n## is the set ##I_m##. So, the function would be one to one and onto and hence we have ##A \thicksim I_m##. Which proves that ##A## is a finite set. Furthermore, if ##A \ne I_n##, then ##A## is a strict subset of ##I_n## and ##A## would have less number of elements than ##I_n##. So, we have ##|A| < n##. Is the proof valid ?

Thanks