The source said that Khloe had everything she could ask for in her day-to-day life in Cleveland, and said she her transition into motherhood had gone "very smoothly".
In the short video, captioned: "Happy One Month True!", the little tot can be seen smiling up at her mum as she looks intently at the camera, gently rocking side-to-side as she posed for the camera.
Khloe Kardashian was feeling "loved" on her first Mother's Day as a mom.
"Picking a name for your child is really so tough", she said in her most recent post.
Five different women came forward with stories about hooking up with the athlete last month and DailyMail.com was the first to break the news of his infidelities.
"Her daughter is Black which proves that O.J. Simpson is Khloé's dad", one person commented. "Tristan and True are twins lol it's insane", she wrote when asked by a fan who True looked like more.
Kardashian replied, "I mean... her dad is black silly. It's been hard for Khloé, but she is doing what she believes is best for her family", the source added.
"She doesn't care what anyone thinks", they continued.
"Khloé is chose to point out that she is completely satisfied in Cleveland", the supply informed PEOPLE.
"She has help so she can leave for shorter workouts, but otherwise she spends most of her time with the baby", the source says. "She is completely satisfied about her life". But that doesn't mean she's lost her sense of humour, which we know from the weird / awesome present she gave her sister Kim for Mother's Day.
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