the title might be a bit absurd but this is how I currently feel just because I can't see where my reasoning is faulty.
So the lecturer presented the lemma as the following:
Let $M \in C[a, b]$. Then $\int_{a}^{b} M \cdot h \,dx = 0, \forall h \in C_0^1[a,b] \implies M \equiv 0$.
But how does the lemma still hold when $h(x) = 0$? The integral would still be zero but it would not imply that $M = 0$ since it could for example be $M = x$. Am I wrong to believe that $0 \in C_0^1[a,b]$?
He gives a proof by contradiction using a non-zero $h$ (the bump function ) somewhere in the lines of: Say $M$ is greater than zero and $h$ is greater than zero, then the integral must also be greater than zero which results in a contradiction since the integral must be equal to zero. Therefore $M$ must be zero. This is obviously true for the case when $h$ is non-zero but isn't this an obvious loss of generality? When $h$ is also allowed to be zero then one can not easily deduce which function is making the integral to be zero. If the lemma would require that it applies to $\forall h \in \{C_0^1[a,b] \setminus 0\}$ then everything would make sense but that is not how the lemma is generally expressed.
Any help would be appreciated.
You seem to be reading it as$$\color{red}{\forall h\in C_0^1[a,\,b]\left(\int_a^bMhdx=0\implies M=0\right)},$$but it actually means$$\color{limegreen}{\left[\forall h\in C_0^1[a,\,b]\left(\int_a^bMhdx=0\right)\right]\implies M=0.}$$It's the equivalent of confusing$$\color{limegreen}{\text{If all people like me, I'm popular}}$$with$$\color{red}{\text{If you pick any one person, if they like me I'm popular}}.$$