I want to show the following inequality regarding schur's majorization theorem $$\overset{n}{\underset{i = 1}{\sum}}a_i\lambda_i\leq \overset{k}{\underset{i = 1}{\sum}}\lambda_i,$$ where $\lambda_1\geq \lambda_2 \geq \dots \geq \lambda_n =0,~ 0\leq a_i\leq 1$ and $\overset{n}{\underset{i = 1}{\sum}}a_i =k$.
Any help will be appreciated.
Extra information: here $a_i (i =1\dots n) $ are diagonal entries of some real symmetric matrix whose eigenvalues are $1$ (with multiplicity $k$) and $0$ (with multiplicity $n-k$).