# Proving correctness of LR parser facts

I have came across following facts while reading some compilers related text. However I did not find them in any standard reference book (mainly dragon book). Are they correct? If yes, how can we prove them?

Fact:

1. If there is a λ-free LL(1) grammar for a language, then we can also prepare SLR(1) grammar for it. (λ-free means: there is no null productions of the form $$A\rightarrow\lambda$$ for any non terminal $$A$$)
2. LL(1) grammar whose variable are all able to derive a not null string is LALR(1). (able to derive a not null string means: there may exist null production $$A\rightarrow \lambda$$, but along with it, there should also exist non-null production $$A\rightarrow \alpha$$ for every such $$A$$, where $$\alpha = (V+t)^*$$ where $$V$$ is any non terminal and $$t$$ is any terminal)