If the nonce is 32 bits why the proof of it is difficult? I want to know why mining of nonce is difficult? 

The nonce is 32 bits (4294967296) and a high level Nvidia graphic card can check and mine it in a second.

why do we need a pool to mine a blockchain?
The vast majority of the time, none of the 2^32 possible nonces in a particular block header template will result in the header having a hash that meets the proof-of-work target.  In fact, the probability that this doesn't happen equals exactly 1 over the network difficulty.  Right now, that is 1/6,379,265,451,411.

When no nonce succeeds, one must change something else in the block header and start again.  The typical solution is to change the extranonce field in the coinbase transaction and recompute the Merkle root.  See What is the extraNonce?