Mobile wallets don't store the data for the entire blockchain. How can you confirm that a transfer really made it into a block?
To prove that a transaction is in a particular block, you only need a few hashes along the path from that transaction to the Merkle root. The number grows logarithmically with the number of transactions.
On platforms that provide proof of reserves, you can follow the official instructions to manually verify your own balance using a Merkle path.
The header of every $BTC block contains a Merkle root that summarizes all the transactions in that block. Currently 70560.
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