We know that a rise in real interest rates will cool down the economy in terms of investments. An increase in interest rate will provide higher incentive for saving rather than consumption. So people will start to save in banks and banks would be having a high stock of these savings. Now, effectively this stock should compel banks to lend more to the people needing it and hence it should drive down interest rates and raise investment levels. How is this fallacy arising ? Please explain
You are getting confused because you are "reasoning from a price change." Interest rate changes don't just happen by themselves, so they cannot be the cause of other effects in the economy.
What actually happens is:
The central bank drains money from the economy.
The interest rate has to rise, otherwise people would try to borrow more money than there is available.
The fact that other people can step in and become lenders may mean that the real interest rate wouldn't rise very much. But the central bank can take this into account. It can just drain extra money from the system to achieve its interest rate target.
The price and quantity of a scarce resource are determined by supply and demand. The quantity which would be offered at some imagined price increases with that price. The quantity which would be purchased at some given price decreases with price. The actual quantity and price of every transaction will then naturally be the level where the quantity offered equals the quantity demanded.
In the case of savings and investment, the price is the interest rate and the quantity is the amount lent and borrowed.
So there is no fallacy. The interest rate is the one agreed to when a lender (saver) and a borrower make a deal. The interest rate isn't the cause, it is the result of the agreement. The interest rate bid and offered at a given moment are two different numbers. There is no fallacy because there is no transaction. At the moment of transaction, there is a single number: the "interest rate" itself. So again, no fallacy.