Editor's Note: This is the first in a continuing series providing snapshots of how fiscal challenges are affecting city government.   Mayor Dave Eggers thanked former city employees for their work Monday before diving into the most restrictive budget in recent history. It is down $3.8 million from last year; five people lost a city paycheck because of cuts. "Our prayers are with you and your families," Eggers said. "It is indeed a challenging time." Several weeks before, City Manager Rob DiSpirito projected a $1.2 million shortfall, but was able to present a balanced budget Monday because of …