Proof that Recall ≥ F1-score I need to prove mathematically that the value of Recall is always bigger or equal to the F1-score.
The equation would be: $2 \times Precision \times \frac{Recall}{(Precision + Recall)} \le Recall$
Does someone have an idea how I can prove this mathematically?

It isn't in general. R code:
> precision <- 0.9
> recall <- 0.5
> 2*precision*recall/(precision+recall)
[1] 0.6428571

A possible confusion matrix giving rise to these precision and recall values would be:
          Predicted P   Predicted N
Actual P       9             9
Actual N       1             5

Note that recall and precision suffer from precisely the same problems as accuracy.
