how to update Safari browser?

You need to upgrade the computer's OS or switch to a product such as Firefox.

To upgrade, open the Mac App Store and try downloading Yosemite. If you get told it's incompatible, choose About this Mac from the Apple menu, check if the computer has at least a Xeon or Core 2 Duo(not Core Duo) CPU and 2GB of RAM, and if it does, click here and order a download code for Lion 10.7.

Back up your data and check your applications for compatibility before upgrading. In particular, Mac OS X 10.7 and newer don't support PowerPC programs such as versions of Microsoft Office prior to 2008.

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I have just bought my macbook in the restore shop.

It has Mac OS X version 10.6.8. If i run software update there are no updates. But still, when i open my gmail mailbox i get notification to update my safari... but there are no updates... what to do?

What I wrote above.

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Firefox