"Ok, so then basically what I make out from what you're saying is that as the dynamic pressure remains constant if we keep a same airspeed as we climb, the anemometer senses the change in density (reduction) and so it is the anemometer that calculates the increased v and displays it in knots on the Airspeed indicator right?"
No.
The Airspeed Indicator has no means of measuring air density.
It simply measures the dynamic pressure and gives you an indicated airspeed that is determined by that pressure.
If the air density is equal to that at ISA msl, then the indicated airspeed will equal TAS (ignoring instrument errors and pressure sensing errors).
But if the density is less than ISA msl, then the indicated airspeed will be less than TAS.
If the density is higher than ISA msl then the indicated airspeed will be more than TAS.
But the Aispeed Indicator has no way of measuring density or TAS, and it does not actually indicate TAS.