The Government will prove the likely source in its case in chief. But, then to really nail the point, if the defendant testifies in his own defense, the Government can ask the defendant about any other sources. If the defendant can't come up with them, well ...
In United States v. Dorsey, 2012 U.S. App. LEXIS 19978 (3rd Cir. 2012), here, a nonprecedential decision, the defendants, husband and wife, were convicted of "conspiracy to defraud the IRS in violation of 18 U.S.C. § 371, filing false tax returns for tax years 2004 through 2006 in violation of 26 U.S.C. § 7206(1), and, as to Troy Dorsey only, structuring cash deposits to avoid tax reporting requirements in violation of 31 U.S.C. § 5324(a)(3) and (d)(2)."
The Third Circuit affirmed on appeal, noting in footnote 3:
Although it was not part of the government's affirmative case, we note that Troy Dorsey testified at trial. When asked, he could not identify the amount of cash he deposited over the three-year period in question, and he did not offer any alternative, non-taxable explanation of its source.Perhaps another reason not to testify at trial in these types of cases.
Note also that the burden on the Government to disprove (at least to some extent) is the reason that, during a criminal investigation, defendant's counsel may throw out leads that the Government is required to track down in order to make the required proof at trial. If the source is members overseas -- e.g., family members in Iran -- that may be a difficult burden to overcome. But, the lead may also lead the Government to more serious money laundering or Bank Secrecy Act charges.