/sci/ - Science & Math

What is the exact meaning of quotient spaces? i mean i know the definition, but what is the essence of it? take [math]\mathbb{R} / \mathbb{Z}[/math] for example.

What book do you recommend as a gift for a friend who is a freshman mathfag?

Hi /sci/, physicsfag here. Is it worth it to get a second degree in mathematics or should i just proceed to a masters degree on mathematical physics (i.e. QFT)? I mean i really like them and been studying math all the time. Not baby math too, currently studying deRham cohomology and Lie theory.

Let A be a non empty subset of [math]\mathbb{R}[/math]. Suppose that there exists a sequence of real numbers [math]x_n[/math] such that:
i) every term of [math]x_n[/math] is an upper bound of A
ii) [math]x_n \rightarrow a[/math] for some [math]a \in A[/math].

prove that [math]a=max\{A \}[/math].