You have your covariance matrix which is all positive, and for the first principal component you are going to solve an eigen-decomposition problem on your covariance matrix (i.e. find the eigen-vectors and eigen-values and retain the one with the highest eigen-value since this corresponds to the variance of each component). You don't have orthogonality conditions on the first component which means you are only interested in getting the one with the highest variance.
So what this boils down to is proving that all elements in the eigen-vector corresponding to the highest eigen-value are all positive (since the covariance matrix is positive definite, it will always have positive eigenvalues).
As for the next one, this lies on the argument dealing with orthogonality since all later components will always be orthogonal to every other principal compnent.
The easiest thing to do is that if the first component is <a,b,c,d,....> then if something is normal you know that X = PC1 Y = PCN then <X,Y> = 0 and the only way for PCN to have this relation is for at least one of the components of PCN to be negative (we also assume PCN is not the zero vector and has positive magnitude).