Let $\lambda=(\lambda_1,\dots,\lambda_r)$ is a partition of $n$ and denote $\lambda'$ the conjugate partition of $\lambda$, with $\lambda'_j=\#\{i\,|\,\lambda_i\ge j\}$.
I'm struggling to try to prove that $(\lambda')'=\lambda$ using only the definition of conjugate partition and without using Ferrers diagrams.
More specifically, I'm trying to justify why these two numbers are equal $$(\lambda')'_k=\#\{j\,|\,\lambda_j'\ge k\}=\#\{j\,|\,\#\{i\,|\,\lambda_i\ge j\}\ge k\}$$
$$\lambda_k=\#\{j\,|\,\lambda_k\ge j\}$$
but I can't figure it out with the indexes and sums appearing.
Could someone help me to develop the steps?