The following statement is a well-known lemma of Ramsey number. $$R(m+1,n+1) \leq {m+n \choose m}$$
Now, I want to prove the improvement of the above statement: $$R(n_1+1,n_2+1,\cdots,n_k+1) \leq{n_1+n_2+\cdots+n_k \choose n_1,n_2,\cdots,n_k}$$
I found some references mentioning it but they just used it as an obvious 'fact'.
I tried to use an induction on $k$ or use the following lemma,
$$R(p_1,\cdots,p_{c-1},p_c) \leq R(p_1,\cdots,R(p_{c-1},p_c))$$
but unfortunately, both strategies did not work well.
Would you help me?
Thanks.