1. closed interval

Hello : I have a problem whit this exercise

Decide which of the following are subrings of the ring of all functions from the closed interval [0,1] to
$\mathbb{R}$:

1)the set of all rational linear combinations of the functions sin (nx) and cos (x) where m, n
$\in{}$ {0,1,2,...}

2)
the set of all polynomial functions

3)the set of all functions that have an infinite number of zeros

2 is a subring but not 1 and 3, but I can not establish the conclusion good, I hope some help

Thanks

2. Re: closed interval

What conditions have be satisfied to be a subring?

3. Re: closed interval

a subring S of a ring R is a subset that is closed under addition and multiplication in the ring, and obviously must be different from the empty set

4. Re: closed interval

for 3) can you think of a function with an infinite amount of 0's, and another one with an infinite amount of 0's then when added give a function with a finite amount of 0's?