Is HDMI sufficient for LG 25UM58?

Officially, the HDMI port (HDMI 1.1 I think) on your MacBook Pro model will max out at 1920×1200 resolution, so less than the native resolution of that monitor.

The two Thunderbolt 2 ports have far more capacity. But then you would need a Thunderbolt 2-to-HDMI adapter, one that outputs more than 1920×1080 with a good frame rate. E.g. the Kanex adapter is Thunderbolt 2 compatible on one end and supports up to 4K on the HDMI side (HDMI 1.4). You will still need a HDMI cable.

That should work because when you plug in a Mini DisplayPort Device, (which is a subset of all the signals required fro ThunderBolt) the computer provides Mini DisplayPort, and the converter converts it to HDMI.

I did not mean to suggest this was the best way to go at all. I meant to suggest you should pay more and get a good native DisplayPort Display, instead of gambling on what might work (or might not) with an HDMI-only display. If it works only at 30Hz, it flashes (MUCH worse than UK TV sets) and you are stuck with it.

Mini DisplayPort is not the same as Thunderbolt or Thunderbolt 2, although the plug has the same shape. E.g. you can’t use a Thunderbolt cable to hook up a Mini DisplayPort screen, or you can’t use the Mini DisplayPort connection on a Mac to hook up Thunderbolt devices. This article seems to suggest it should work, though. Confusing stuff.

Thunderbolt: Connect a Thunderbolt display, or connect a Mini DisplayPort to DVI, VGA, HDMI, or DisplayPort adapter. You can also connect storage devices, video capture, FireWire, and gigabit ethernet adapters directly to the Thunderbolt port on your Mac.

HDMI was invented for HD TV sets running at 1920 by 1080p.

It can sometimes be pushed farther, by selecting exactly the correct settings in both the computer AND on-screen display settings, and using cables explicitly stating they are for higher data rates (not just 'high quality'). Unfortunately, a lot of this HDMI extension is Black Magic, and if it fails to work or drops back to 30Hz refresh rate, there is nothing to be done for it.

The best modern displays are using DisplayPort interface. it has higher data rates and fewer problems, and the screen contains it own internal Screen Buffer. It also use a a packet-like interface, and does not require the 60 Hz "heartbeat" refresh required by "legacy" displays. The link goes mostly quite when the screen stops changing. This means the Hardware at both ends runs cooler and lasts longer, and it uses less battery power on a Macbook Pro.