Based on that, the algebra calculation is correct. But it seems confused to me. Even though the overall efficiency reduces, it is still useful, isn't it? From the same amount of energy we can produce more work. So even though the efficiency reduces , it is still energy saving. Why don't people use this method?
Even though the overall efficiency reduces...
The efficiency doesn't "reduce". It just doesn't increase as much as one might, at first, hope.
Why don't people use this method?
It is widely used. When two heat engines are used this way, it is called a "combined cycle".
You can, for example, buy a combined-cycle power plant from General Electric. GE explains that the advantage of its combined-cycle plant is higher efficiency:
A combined-cycle power plant uses both a gas and a steam turbine together to produce up to 50 percent more electricity from the same fuel than a traditional simple-cycle plant.
This shows the design of a Nooter Erickson combined cycle plant:
Combustion of fuel and air drive a turbine (the upper one in the diagram). Heat from the exhaust of that turbine is captured to drive the steam turbine (the lower one in the diagram).