[Doc] Update reasoning with stream example to use OpenAI library (#14077)

Signed-off-by: liuyanyi <wolfsonliu@163.com>

docs/source/features/reasoning_outputs.md

@@ -78,7 +78,55 @@ Streaming chat completions are also supported for reasoning models. The `reasoni
 }
 ```
 
-Please note that it is not compatible with the OpenAI Python client library. You can use the `requests` library to make streaming requests. You could checkout the [example](https://github.com/vllm-project/vllm/blob/main/examples/online_serving/openai_chat_completion_with_reasoning_streaming.py).
+OpenAI Python client library does not officially support `reasoning_content` attribute for streaming output. But the client support extra attributes in the response. You can use `hasattr` to check if the `reasoning_content` attribute is present in the response. For example:
+
+```python
+from openai import OpenAI
+
+# Modify OpenAI's API key and API base to use vLLM's API server.
+openai_api_key = "EMPTY"
+openai_api_base = "http://localhost:8000/v1"
+
+client = OpenAI(
+    api_key=openai_api_key,
+    base_url=openai_api_base,
+)
+
+models = client.models.list()
+model = models.data[0].id
+
+messages = [{"role": "user", "content": "9.11 and 9.8, which is greater?"}]
+stream = client.chat.completions.create(model=model,
+                                        messages=messages,
+                                        stream=True)
+
+print("client: Start streaming chat completions...")
+printed_reasoning_content = False
+printed_content = False
+
+for chunk in stream:
+    reasoning_content = None
+    content = None
+    # Check the content is reasoning_content or content
+    if hasattr(chunk.choices[0].delta, "reasoning_content"):
+        reasoning_content = chunk.choices[0].delta.reasoning_content
+    elif hasattr(chunk.choices[0].delta, "content"):
+        content = chunk.choices[0].delta.content
+
+    if reasoning_content is not None:
+        if not printed_reasoning_content:
+            printed_reasoning_content = True
+            print("reasoning_content:", end="", flush=True)
+        print(reasoning_content, end="", flush=True)
+    elif content is not None:
+        if not printed_content:
+            printed_content = True
+            print("\ncontent:", end="", flush=True)
+        # Extract and print the content
+        print(content, end="", flush=True)
+```
+
+Remember to check whether the `reasoning_content` exists in the response before accessing it. You could checkout the [example](https://github.com/vllm-project/vllm/blob/main/examples/online_serving/openai_chat_completion_with_reasoning_streaming.py).
 
 ## Structured output