Is it bid’ah for a person who is already there to say salaam to one who has just arrived?

 

With regards to greeting salaam. I have notice some muslim people don't greet salaam when they arrive so it prompted me to greet them salaam which is the opposite of what should be. My question is am I doing bidaa because of this (greeting salaam to the person who arrives who does not say salaam).. did I sin and second thing should I greet next time with salaam a person who just arrive with he / she didn't say salaam.

Praise be to Allaah.

The Sunnah is for one who is walking to greet one who is
sitting, and for one who is riding to great one who is walking, and for the
younger to greet the older, and for one who is coming in to greet the people
who are in a place, because Allaah says (interpretation of the meaning): 

“But when you enter the houses, greet one another with a
greeting from Allaah (i.e. say: As‑Salaamu ‘Alaykum — peace be on you),
blessed and good”

[al-Noor 24:61] 

And the Prophet
(peace and blessings of Allaah be upon him) said: “The one who is riding
should greet the one who is walking, and the one who is walking should greet
the one who is sitting, and the smaller group should greet the larger
group.” Al-Bukhaari, 6234; Muslim, 2160. According to a report narrated by
al-Bukhaari: “One who is passing should greet one who is sitting.” 

It is well known that initiating the greeting of salaam is
Sunnah and is mustahabb (encouraged), and responding is waajib
(obligatory). 

If the one who is coming in does not say salaam, but the one
who is in the house does say it, or if the one who is walking does not say
salaam but the one who is sitting does say it, there is nothing wrong with
that, rather he has done something good and has fulfilled the Sunnah of
salaam, and the other one is obliged to respond. 

Al-Nawawi (may Allaah have mercy on him) said: Note that
initiating the greeting of salaam is Sunnah and mustahabb, but not
obligatory. It is Sunnah on the basis of kafaayah (a communal Sunnah). If it
is a group that is giving the greeting, it is sufficient for one of them to
say the salaam, but if all of them say salaam, that is better… As for
returning the greeting, if it is one person that is greeted, he is obliged,
on an individual basis, to return the greeting, but if it is a group then
responding to the greeting is a communal obligation (fard kafaayah) for
them; if one of them responds, then there is no blame on the others, but if
all of them fail to respond, then all of them have sinned. If they all
return the greeting, this is best. This is the view of our companions and it
is clear and good. End quote from al-Adhkaar, p. 356 

Then he (may Allaah have mercy on him) said: Chapter on
etiquette and issues of greeting with salaam. It is narrated in Saheeh
al-Bukhaari and Saheeh Muslim that Abu Hurayrah (may
Allaah be pleased with him) said: The Messenger of Allaah (peace and
blessings of Allaah be upon him) said: “The one who is riding should greet
the one who is walking, the one who is walking should greet the one who is
sitting, and the smaller group should greet the larger group.” According to
a report narrated by al-Bukhaari: “The younger should greet the older, and
the one who is walking should greet the one who is sitting, and the smaller
group should greet the larger group.” 

Our companions and other scholars said: What is mentioned
here is the Sunnah, but if they do it differently and one who is walking
greets one who is riding or one who is sitting greets one who is riding or
walking, that is not makrooh. This was clearly stated by Imam Abu Sa’d
al-Mutawalli and others. Based on this, it is not makrooh for a large group
to greet a smaller group first, or for an older person to greet a younger
person first. End quote from al-Adhkaar, p. 369 

Al-Haafiz ibn Hajar narrated that al-Maaziri said: If one who
is walking initiates the greeting and says salaam to one who is riding, that
does not mean that he is not following the command to spread the greeting of
salaam, but paying attention to what is proven in the hadeeth is better. It
is mustahabb to give the greeting in the manner prescribed, but not doing
what is mustahabb is not necessarily makrooh, rather it is contrary to what
is best. If the one who is enjoined to initiate the greeting fails to do so,
and the other one initiates it, then the one who was enjoined has failed to
do something mustahabb and the other one has done something that is Sunnah
unless he hastened to do it, in which case he has also failed to do
something that is mustahabb. End quote from Fath al-Baari, 11/17 

The phrase “unless he
hastened to do it, in which case he has also failed to do something that is
mustahabb” means that the one who is in the house, for example, should not
hasten to say salaams to one who is coming in, rather he should give him
time to say the salaam, then if he does not do so, the one who is in the
house should greet him with salaam.

 Conclusion: If you say salaam to the one who enters upon you
but does not say salaam, this is not a bid’ah or a sin, rather it is
reviving the Sunnah and spreading love, so long as you do not hasten to do
that, rather you should give the one who is coming in time to greet you with
salaam, then if he does not do that, you should say the salaam in that
case. 

And Allaah knows best.