褒める and 励ます: praising merits and offering encouragement
Compare praising careful writing or effort with encouraging a dejected friend or a runner. Preserve the overlap between praise and encouragement without adding recovery or a completed race.
Praising a result and the effort behind it
褒める（ほめる） means to praise. 先生は丁寧に書けた字を褒めた。 says that the teacher praised the carefully written characters. Identify the merit being recognized.
結果だけでなく努力した過程も褒める。 says that praise concerns not only the result but also the effort that went into it. Read だけでなく and も together to preserve the additional target.
Encouraging a friend or a runner
励ます（はげます） means to encourage. 落ち込んだ友人を言葉で励ました。 describes encouraging a dejected friend with words. It names the friend's situation and the means of encouragement.
観客の声援が走者を励ます。 says that the crowd's cheers encourage the runner. The runner receives the encouragement, but a finish or victory is not stated.
Praise can also encourage
Praising someone can provide encouragement, so the purposes can overlap. Do not make 褒める always a past assessment and 励ます always encouragement about the future.
The examples compare recognized merits in writing or effort with the situation of someone being encouraged. Encouragement itself does not establish the friend's recovery or the runner's result.
Does 観客の声援が走者を励ます。 establish that the runner finished the race?
Show the answer and why
No. It says that the crowd's cheers encourage the runner. Finishing or winning would be separate information.