# why does lasso select at most n predictors?

From the seminal paper on elastic net regularization from Zou and Hastie 2005, I read

For this kind of
p>>n and grouped variables situation, the lasso
is not the ideal method, because it can only select at most
n variables out of p candidates (Efron et al., 2004).


However, in Efron et al., 2004 I can not fin the proof/demonstration? Any hint?

Consider a linear model $$Y = X\beta + \varepsilon$$ with $$p$$ variables and $$n$$ observations, $$p>n$$. Assuming the variables are not linear dependent, i.e. the matrix $$X$$ has rank $$n$$, $$Y$$ can be perfectly predictet ($$Y = \hat{Y}$$) using only $$n$$ variables. So LASSO will ideally choose the $$n$$ variables such that $$\lambda ||\beta||_1$$ is minimal. This solution should be unique (because of linear independence on the individual variable level) and threrefore, all perfect fits of the linear model that include more than $$n$$ variables will have a higher $$\lambda ||\beta||_1$$ and are therefore not optimal.