These are the extracts from Vishnu Purana written in BC.
ततश्च भारतं वर्षमेतल्लोकेषुगीयते
भरताय यत: पित्रा दत्तं प्रतिष्ठिता वनम (विष्णु पुराण, २,१,३२
This country is known as Bharatavarsha since the times the father entrusted the kingdom to the son Bharata and he himself went to the forest for ascetic practices
Vishnu Purana (2,1,32)
uttaraṃ yatsamudrasya himādreścaiva dakṣiṇam
varṣaṃ tadbhārataṃ nāma bhāratī yatra santatiḥ
उत्तरं यत्समुद्रस्य हिमाद्रेश्चैव दक्षिणम् ।
वर्षं तद् भारतं नाम भारती यत्र संततिः ।।
"The country (varṣam) that lies north of the ocean and south of the snowy mountains is called Bhāratam; there dwell the descendants of Bharata."
- Vishnu Purana (2.3.1)
This again proves you wrong....