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

                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 Allah.
    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.