Lower bound on sum of $k$ largest entries

Question

I know that this question may be easy, but I need this result (if it is true) in a large proof for a problem:

The Question is: for any set of positive real numbers $\{x_1,x_2,\ldots,x_n\}$ and assume that it is in descending order then:

$$\sum_{\ell=1}^k x_\ell \geq\frac k n \sum_{\ell=1}^n x_\ell$$

for $k=1$, we know that it is true since: $x_1 \geq\frac 1 n \sum_{\ell=1}^n x_\ell$ (maximum value is greater than the average)

Is the above inequality true for any $k$?

I appreciate your answers. Thank you

Answer 1

Dividing both sides by $k$ yields this: $$ \frac 1 k \sum_{\ell=1}^k x_\ell \geq\frac 1 n \sum_{\ell=1}^n x_\ell. $$ That says the average of the first $k$ numbers is at least as big as the average of the all of them.