A 0-0 draw at home to Sporting CP on Wednesday saw Arsenal win their quarterfinal tie 1-0 on aggregate.
Borges reflected afterwards, "I think 'proud' is the right word for all the players, for what they've done over these two games. I think we deserved more, potentially extra time. In both games, Sporting had the better chances. The character and the personality from the team, to fight against one of the best teams in Europe, was amazing.
"In terms of strategy, the team was great. We had the better opportunities, Arsenal didn't create much in our box, neither here nor in Lisbon. We were great on the ball and we didn't allow a team with great individual players who can resolve a game one-against-one to break.
"Sporting are a team that likes to have the ball, play offensively and have possession. We didn't shy away from that. We showed our character and our personality.
"There can't be any frustration, just pride at what they've managed: two big games against one of the best teams in Europe. The word 'frustration' does not exist for us, just recovery. If we want to be among the best in Europe, we have to have the mental capacity to respond to having a game every three days. That's extra motivation for us."
Borges added: "In terms of a summary for Sporting in the Champions League, it's been very positive. We've showed the history of Sporting. We can lift our heads now and look at all the positive things we've done.
"It was the start of a dream and the result of growth that the club has been doing at all different levels. We've created this respect that people look at Sporting and also the Portuguese league differently."