Sono venuto a vedere se stai/stavi bene

Question

I have read the following sentence in an Italian learning tool:

"Sono venuto a vedere se stai bene"

Wouldn't the correct be "Sono venuto a vedere se stavi bene", since "sono venuto" is a past tense?

Answer 1

"Sono venuto a vedere se stai bene"

While speaking, a person doesn't think too much about what grammar says, and often the speaker makes errors. Often these errors become so frequent that dictionaries, after some time, record them as correct. First of all, hence, as this phrase sounds normal, it is normal and so it is correct (this idea is debatable, but not totally wrong).

But in this case there is something more because, as already explained in comments, this phrase contains two distinct concepts: 1) "Sono venuto (a vedere)", 2) "Se stai/stavi bene". The first part:

1) Sono venuto a vedere... (I came here to see/check...)

uses the past because the action is already terminated. Because of this, it's simply more natural to use the past than to use the present, which instead would be used, for example, in a immediately preceding telephone call: "Allora vengo a vedere" (then I come to see). The choice is up to the speaker.

Next:

2) (a vedere) ...se stai bene (...whether you are fine)

is just the motivation of the visit, a totally separate concept which does not need to match with the previous one. But note that the speaker could also say

2) ...se hai mangiato (...whether you did eat something)

or even use the future. Maybe (this is true for me, for example), the speaker uses the past because in its mind it connects the care about the other person with the act of moving on to the other person. With the two ideas connected, at the time of the visit both are placed in the past:

"Sono venuto a vedere se stavi bene": that past "stavi" can be there because, in the mind of the speaker, the time of the decision to visit is back in the past.

Consider also that the imperfect can be used to add courtesy (Imperfetto di cortesia: https://it.wikipedia.org/wiki/Indicativo_imperfetto#Imperfetto_di_modestia_o_di_cortesia); a speaker can choose to not use the imperfect just to avoid the courtesy.