Show that $\cos t,\,\cos^2 t,\, . . . $ is a linearly independent system in $F$ I'm faced with this problem which I can't really know how to begin solving it, any help would be appreciated.

Let $F$ be the real vector space of the functions $f : \Bbb R\to\Bbb R$.
  Show that: $$\cos t,\; \cos^2 t,\; . . . ,\; \cos^n t,\; . . .$$
  is a linearly independent system in $F$.  

Any guidance is welcome.
 A: To prove that an infinite set is linearly independent it is sufficient to prove that every finite subset of it is linearly independent.
So show that $\{\cos t,\cos^2t,\cos^3 t,....\cos ^n t \}$ linearly independent.
For this note that :$c_1\cos t +...+c_n\cos ^n t=0$ .Putting $t=0$ we have  $c_1+c_2+....+c_n=0$.
Now differentiating w.r.t we get $c_1\sin t+c_2 \sin 2t+...+c_n n\cos^{n-1}t \sin t=0$.Putting $t=\frac{\pi}{2}$ we get $c_1=0$.
In this way go on differentiating and substitute suitable values (try for $n=3$ first) , you will get $c_i=0\forall i$
A: So you start with the equation
$$\sum_{k = 1}^n \lambda_k \cos^k t = 0.$$
Now consider what this equation means: the equality must be true for each $t\in \mathbb{R}$...
A: Start from required $$\sum_{k = 1}^{\infty} a_k \cos^k t = 0 \;\;\;\; (1)$$
and now divide through $\cos(t)$. (If first few $a_{i}, i < m$ are $0$ reduce to the case when first coefficient is different from $0$ simply by dividing by $\cos^{m-1}(t)$.)  You have got:
$$a_{1}+\sum_{k = 1}^{\infty} a_{k+1} \cos^k t = 0$$ or
$$\sum_{k = 1}^{\infty} a_{k+1} \cos^k t = -a_{1} \neq 0 \;\;\;\; (2)$$
We can bound $t$ to be from $0$ to $\frac{\pi}{2}$. It is irrelevant if we include or exclude one of or both values. Now
$$\lim\limits_{t \to 0^{+}} \sum_{k = 1}^{\infty} a_{k+1} \cos^k t = -a_{1} \;\;\;\;$$ which means that 
$$\sum_{k = 1}^{\infty} a_{k+1}$$ is bounded since  $\lim\limits_{t \to 0^{+}} \cos^k t = 1$. This means that since $\lim\limits_{t \to \frac{\pi}{2}^{-}} \cos^k t = 0 $ it must be
$$\lim\limits_{t \to \frac{\pi}{2}^{-}} \sum_{k = 1}^{\infty} a_{k+1} \cos^k t = 0$$  contrary to our assumption that it is $-a_{1} \neq 0$.
This means that for all $i$ $a_{i}=0$ so the set of functions is independent.
