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BTC lite-wallet sends a bloom-filer request to bitcoin node about what is the current balance of the particular address. After that wallet will have a Merkle-tree prove that transaction is indeed on a blockchain.

The question Do lite-wallets store all headers since genesis block? A block header is just 80 bytes, however, 80 bytes *6 blocks/hour *24h*365d*9y ~37M which is a lot for a lite wallet.

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No, lite wallets can function just fine without storing all the block headers. Instead, what wallets usually do is confirm if there is a new tx involving one of the wallet addresses, and then download that specific tx. They will then only store the tx, and not the rest of the information.

As their UTXO set expands, they will check new blocks to see if they contain a new UTXO, or consume an existing UTXO that the wallet cares about.

Additionally, many wallets these days keep track of when an address was created, and will not scan blocks before that.

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