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Help! let G be a finitely generated group and H, a subgroup of G. If the index of H in G is finite, show that H is also finitely generated

Math competition in SerbiaCan somebody solve this, sorry for bad translation... thanks a lot

Math Problem: In a movie theater, which has 2015 seats came in 2014 visitors, including the Mika. All these visitors are moved to an arbitrary places, ignoring the fact that place their scheduled according to the ticket. to half of the film enters 2015 visitor. he just wants to sit in their place according to the ticket, and if it is busy, he will raise the viewer from that place. Raised viewer then looks his seat, and if it is busyhe will raise the viewer sitting in his place. This process continues until there is lifted a viewer who is assigned to the ticket place that is free (and then he will sit on the site). Let Abe, the number of initial arrangement that will throughout this commotion and Mika in a certain point will be raised, and B of the remaining initial schedule. Which of the following three relationships apply: a>b, a=b or a<b?

Math Problem from competition in Serbia. Can somebody solve this, sorry for bad translation... thanks a lot

Math Problem: In a movie theater, which has 2015 seats came in 2014 visitors, including the Mika. All these visitors are moved to an arbitrary places, ignoring the fact that place their scheduled according to the ticket. to half of the film enters 2015 visitor. he just wants to sit in their place according to the ticket, and if it is busy, he will raise the viewer from that place. Raised viewer then looks his seat, and if it is busyhe will raise the viewer sitting in his place. This process continues until there is lifted a viewer who is assigned to the ticket place that is free (and then he will sit on the site). Let Abe, the number of initial arrangement that will throughout this commotion and Mika in a certain point will be raised, and B of the remaining initial schedule. Which of the following three relationships apply: a> b, a = b or a <b?