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During Jesus' ministry on earth, he sent the disciples out in pairs and gave them the power to teach, heal the sick, and cast out demons (Mark 6:7-13). In the first part of Acts 3 we see that the Jesus is continuing to work through His followers by the power of the Holy Spirit. This healing of the lame man demonstrates that the apostles were still acting with Jesus' authority and power, not their own. Just as Jesus used miracles to establish the authority of His teaching, the apostles are able to do the same to demonstrate their authority to speak about Jesus.
One note, this healing is very similar to Jesus' healing of the lame man by the pool (John 5:5-15) but in this case the man had been unable to walk since birth.