Find a Context-Free Grammar for $L:=\{a^nb^mc^{n+m}\mid n,m\in\mathbb{N}\}$
$S\to\lambda$ is unnecessary because $S\to B$ and $B\to\lambda$.
The grammar recognizes the desired language.


You can prove the second assertion pretty easily using induction (for example on the length of the derivation, or the length of the sentence).

You'll probably find it easier to prove two assertions independently:


If a sentence is in $L$, then it is in $L(G)$.
If a sentence is in $L(G)$, then it is in $L$.


Together, these prove equivalence.