Far enough to show that the funds were lawfully obtained by the deceased, using the records that reasonably exist: the estate grant and distribution, and evidence of the deceased's employment, property sales or savings from which the estate was built.
Prove the gift, then prove the giver
Hypothetical example: An inheritance is lawful capital, but USCIS requires evidence of the lawful source in the hands of the person from whom it came. For a Portugal Cove-St. Philip's investor the first layer is straightforward: the will or letters of administration, the estate's accounts, the executor's distribution statement and the deposit into the investor's account.
The second layer is the harder one: how did the parent accumulate the estate? Typical evidence is the sale documents for a family home, pension or employment records, tax filings where available, and bank statements over a period long enough to show the pattern. Perfect records are not expected for a deceased person's lifetime, but unexplained large deposits shortly before death will draw a request for evidence.
The pension lump sum is easier: the plan statement, the commutation election and the transfer into the investor's account, with the tax withheld shown. Every subsequent transfer to the enterprise should be by traceable wire, not cash or third-party accounts.