This February 7, 2010 file photo shows unemployed Americans they pick up job applications from a prospective employer at the National Careers Fair in LA. New US jobless claims fell slightly last week but still remained above 400,000, official data showed on June 2, 2011. Initial claims for unemployment benefits totaled a seasonally adjusted 422,000 in the week ending May 28, a decrease of 6,000 from the prior week, the Labor Department said. The decline was weaker than the average analyst estimate of 413,000 claims.The four-week moving average, which eases volatility, fell by 14,000 claims to 425,500 last week. (File photo / Aol) Aol