Difference (if any) between proof-of-work in mining bitcoin vs verifying transaction I'm trying to understand the proof-of-work idea for verifying a transaction. I think I understand why proof of work is needed to verify transactions (essentially to prevent spoofing/double spending etc). 

But my understanding is this:


the proof of work for verifying a transaction is the same as for mining a new bitcoin
if this is the case, then it becomes harder and harder to verify transactions.


I'm pretty certain the first statement is false, in which case I'd like to know what exactly is the proof-of-work for verifying a transaction.
I think I understand why proof of work is needed to verify
  transactions (essentially to prevent spoofing/double spending etc).


This is right.


  the proof of work for verifying a transaction is the same as for
  mining a new bitcoin


This is right too.


  if this is the case, then it becomes harder and harder to verify
  transactions.


It depends it can become harder as it can become easier. But whatever happens there is no issue with that.

Explanations:

The miner generates blocks which header have the lowest possible sha256 hash until he finds one which sha256 is lower than the Target. This block will be valid and can then be included in the blockchain. Inside of the block the first transaction is different from the others because it hasn't a valid input and it has as transaction output an address controlled by the miner. This address will receive the block reward(new bitcoins that are generated with the block) as well as the transaction fees of
the transactions that were included in the block.

Now actually it's true that generating new blocks become more and more difficult because there are more and more miners with more and more sophisticated mining material.
This is why the target becomes more and more lower in order to maintain an average time between each generated block of about 10minutes.
But if miners will start to stop mining bitcoin the target will increase and mining will become easier again.

This post will help you to understand a little bit better what are the different components of a block.
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