document: Marvyn Iheanacho, 39, attacked Alex Malcolm in Mountsfield Park, Catford, on 20 November 2016.
  Witnesses heard a "child's fearful voice saying 'sorry'", loud banging and a man screaming about the loss of a shoe, Woolwich Crown Court was told.
  Iheanacho, who denied murder, will serve a minimum of 18 years in prison.
  The jury heard Alex suffered head and stomach injuries and died in hospital two days after the beating.
  His mother, Lilya Breha, 30, told the court Iheanacho had also attacked her after she tried to call an ambulance when he returned to her flat carrying her injured son.
  In a tearful interview, she said: "He (Alex) was bubbly. He was just perfect you know, he was a really, really special little boy.
  "He was shy, he was so shy, and very polite. He would always say 'Mummy thank you' and 'I love you Mummy'."
  Ms Breha described Iheanacho as a "good liar" and pathetic.
  Recalling how she met him through a friend after he left prison, she said he had convinced her he was innocent and a good person.
  She said: "When I think about it now, to be honest I feel like it was all such a big lie and he just pretended to be a good guy pretty well."
  Prosecutors revealed there had been "problems with witness interference" during the trial, with Iheanacho phoning Ms Breha from prison to try to persuade her to back him in court.
  In a victim impact statement, Alex's father said he would "never forget seeing him in his hospital bed fighting for his life. That image will stay with me forever."
  "Just thinking about what's happened and trying to put words on paper is tearing me apart," he said.
  Sentencing, Judge Mark Dennis QC said the killer had a deeply entrenched character flaw that "leads you to overreact and lose your temper".
  "You used your undoubted strength and simple brute force," against a "completely defenceless" child, he said.
  He said Iheanacho, who has a string of previous convictions for violent offences, had given fake and misleading accounts to paramedics, hospital staff and police.
summary: A man who battered his girlfriend's five-year-old son to death in a London park for losing a trainer has been jailed for life.