I’m a retail manager and have been for many years.
Your receipt is proof of purchase.
If a receipt is unavailable you can bring in a bank statement. The bank statement will say the date it was sold and the transaction can be easily traced. Every store can do this as different items have different tax codes and they must keep records of the items sold. In most businesses this takes seconds.
If you are exchanging items point out that you are not having a refund and taking money from them. There is absolutely NO INVENTORY LOSS as scaredypants said was the reason for this, so any manager using his common sense would use his discretion and exchange for you.
Proof of purchase can be:
Receipt
Bank statement
Product is company branded (this is harder to prove but trading standards will usually take the side of the consumer)
If all else fails call head office as its very possible you spoke to either a manager who isn’t confident in his own judgement or just plain unhelpful.