That claim is factually wrong on a couple of different levels. First, the closing remarks in Mark 16:9-20 -- the verses that many scholars believe are not part of the original text -- are not the first mention of Jesus' resurrection in the gospel of Mark. Just a few verses before that, there is a discussion of Jesus' resurrection:
He said to them, "Do not be afraid. You seek Jesus of Nazareth, who was crucified. He is risen. He is not here. See the place where they laid him. But go, tell his disciples and Peter that he is going before you to Galilee. There you shall see him, as he told you." (Mark 16:6-7)So if someone were to completely discard the questioned ending of Mark, it would not change the fact that Jesus was plainly stated to have risen in the earlier verses of the gospel of Mark. No matter what your view is about the questioned verses, either way the Gospel of Mark ends with Jesus having been raised from the dead.
Second, many scholars believe some of Paul's letters to be from an earlier date than the Gospel of Mark. It is a mistake to assume that the gospels were written earlier -- that they are the earliest information we have on Christian teaching -- simply because our New Testaments have the gospels collected in the front and the other writings behind. Jesus' resurrection is a key point in Paul's writings. Paul's letter to the Galatians may date to the late 40's or early 50's A.D.; Jesus' resurrection is mentioned in the very first verse of that letter. Paul's first letter to the Thessalonians also may have an early date, in the 50's A.D. Again, Paul does not get far into the letter before he proclaims Jesus' resurrection: "his [God's] Son from heaven, whom he raised from the dead -- Jesus" (1 Thessalonians 1:10). Paul's first letter to Corinth, again often dated to the 50's A.D., contains a lengthy discussion of resurrection and an early list of witnesses who had seen Jesus alive again after he was raised from the dead. Here are texts that many scholars believe to be the earliest Christian writings. Here are texts that most would agree were written in the lifetime of people who knew Jesus in person. And in them, time and again, there are clear references to Jesus' resurrection. There are even references to people still alive in that day who had seen him with their own eyes.
In short, when people say that the Gospel of Mark ended without a resurrection, they are mistaken. They may have confused the closing remarks with the earlier verses where the readers are told of Jesus' resurrection. Or they may not be aware of the earlier verses where the readers are told of Jesus' resurrection. And if the dates that scholars assign to Paul's letters are correct, the announcement of Jesus' resurrection went well back into the lifetimes of those who knew him in person.