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In your case, you have a contractual periodic tenancy and the notice period was agreed in the original AST agreement.
You may therefore insist on your tenant giving 2 months notice as per the contract.
Your tenant may be confusing the tenancy with a statutory periodic tenancy where only one month's notice is required.
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Noted, yes that would be sensible and you may wish to accept the one month's notice and leave it at that.
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