Building Hilbert's Hotel

When he began construction on his Grand Hotel, Dr. Hilbert contacted one of the foremost builders available. The mathematician insisted that the construction begin without delay, and he set a final deadline of 2 months for the project. When the builder complained that there was no way to complete an infinite project in a finite amount of time, Hilbert proposed the following schedule. In the first month, only the first room of the hotel needed to be completed. The builder would be allowed half of the second month for work on the second room. That would leave just over 2 weeks until the opening date. The third room was to be completed in half of the remaining time, giving about 1 week for it's construction. Each successive room was to be built twice as quickly as the preceding one. Correspondingly, the construction of each room would take only half of the remaining time until the grand opening. And finally, despite the builder's increasingly loud complaints, the schedule was adhered to and the hotel opened on time.